.7“ n n. v.,Iv~l “.1 3 LIMA mu ~ VII'yr' ag‘ 3‘ v I of» l : t ‘ ‘ L 4-... -\‘ . ~ ‘1 rim m! a,» 3-er - l‘. ;: v'w‘.‘ . or .f‘/ in 53‘. +" '1 ’1'? . ~ n _H.r 2m." ‘ I rye.“ A; ,. m In x .3: s1; ‘a‘r '5 '4‘ ‘ I ‘. x .o. , MICH GANS TSATE I IIII IIIIIIIIIIIIIIII IIIIIIIIIIIIIIII 31293 00881 2640 III This is to certify that the dissertation entitled Analysis of air flow patterns in potato storage presented by Zai-chun Yang has been accepted towards fulfillment of the requirements for PhD Agri. Engineering degree in Date IgOCfl- q, MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 LIBRARY ‘Mlchlgan State Unherslty PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before data due. DATE DUE DATE DUE DATE DUE T_I ___;__J —I—7 MSU Is An AffirmatIve Action/Equal Opportunity Institution chrnG-nt ANALYSIS or AIR FLOW PATTERNS IN POTATO STORAGE By Zai-dmn Yang A DISSERTATION Submitted to MichiganStateUnivclsIty inpartialfulfillmentoftherequhwements forthedepeeof DOCTOR OF PHILOSOPHY DepartlnentofAu'ictflmralEnglneerlng 1991 ABSTRACT ANALYSIS OF AIR FLOW mums 1N POTATO STORAGE By Zai-chun Yang In a modem potato production system, potatoes are placed in a storage immediately afierhawestmreducelossesmdmpreservequafityforhtermubdngmprwessing. During storage, the quality of potato tubers is strongly affected by the temperature, relative humidity and composition of the air within the potato pile. Therefore, maintenanceofauniformairflowintheductsystemandthroughthestorageis important. To serve this purpose, the effect on air flow patterns of various factors, such as the characteristics of potato tubers, the air properties and air flow parameters, and the configurations of the duct system and the potato storage, Should be thoroughly understood. By taking into consideration the roles of the above factors, general and abstract mathematical models were established based on physical principles and experimental data. Because of their nonlinearity, these mathematical models could not be solved analytically. The finite element method was used to facilitate the solution processes. It was an effective method for dealing with nonlinear partial differential equations in the present study. Following the concepts of the mathematical models and the procedures of applying the finite element method, the computer programs were written in Fortran code. The calculated results included pressure, velocity and air flow direction at any point in two- and threedimensional potato piles, and were plotted as iso-pressure lines, streamlines and velocity profiles. It was found that uniform air flow usually existed in the upper region of the potato pilewherethedepthofthepilewasequaltoorgreaterthanthespacingbetweentwo adjacent ducts. Nonuniform air flow dominated the lower region of the potato pile where the depth of the pile was less than the duct spacing. The region above the lateral duct had reasonable ventilation. But the lower region between two adjacent ducts had the poorest ventilation condition. Here, the pressure showed considerable variation, and the velocity was the lowest at the mid-point between ducts where temperature control would be the most difficult. Duct spacing had a significant effect on air flow patterns in the potato pile. Decreasing duct spacing resulted in more uniform air distribution in the pile and improved the air ventilation conditions in the middle lower region between two adjacent ducts. Duct size had a marked effect on air flow patterns. Increased duct size achieved similar results to decreased duct spacing. A lower potato pile depth was favorable for having more efficient air ventilation and circulation through the pile. Duct pressure had little effect on the iso-pressure lines and the streamlines. The general air ventilation condition in the middle lower region between two adjacent ducts was independent of the duct shapes studied when duets with different shapes have equivalent diameters. However, in-floor rectangular duct tended to have a more uniform air distribution in the pile than ducts with triangular, circular and semicircular shapes. In a potato storage, air flow distribution along the duct axial direction was generally nonuniform. The effect on air flow patterns of the distance from a given cross-section to the duct entrance was equivalent to that of different duct pressures on air flow patterns. TOMYMOTHERLAND-CHINA ACKNOWLEDGMENTS IheauthorwishestoexpresshissincereappreciationtoDr.RogerC.Bmok (Professor and Extension Specialist, Department of Agricultural Engineering) for his encouragement, guidance and support as the major professor and chairman of the doctoral advisory committee. I The author is very grateful to Dr. Larry J. Segerlind (Professor, Department of Agricultural Engineering), Dr. Amritlal M. Dhanak (Professor, Department of Mechanical Engineering) and Dr. Jerry N. Cash (Professor and Extension Specialist, Department of Food Science and Human Nutrition) for their serving on the doctoral advisory committee and for their help and guidance. Special thanks are extended to Dr. Thomas H. Burkhardt (Professor, Department of Agricultural Engineering), Dr. John F. Foss (Professor, Department of Mechanical Engineering) and Dr. Charles A. Petty (Professor, Department of Chemical Engineering) for their help and advice during the author’s graduate study in Michigan State University. The author is indebted to the faculty, staff and students in the Department of Agricultural Engineering for their friendship and help. TABLE OF CONTENTS Page LIST OF TABLES ............................................................................................................ xi LIST OF FIGURES .......................................................................................................... xii LIST OF SYMBOLS ..................................................................................................... xvi Chapter 1 INTRODUCTION ........................................................................................................... 1 1.1 Background ............................................................................................................... 1 1.2 Objectives ................................................................................................................. 3 2 LITERATURE REVIEW ....................................... ..................... 4 2.1 The potato crop ........................................................................................................ 4 2.2 Causes and control of deterioration of stored potato ............................................... 8 2.2.1 Physical problems ........................................................................................... 8 2.2.2 Physiological problems ................................................................................... 9 2.2.2.1 Water loss ........................................................................................... 9 2.2.2.2 Respiration ......................................................................................... 10 2.2.2.3 Sugar level ......................................................................................... 13 2.2.2.4 Sprouting .......................................... - ..... _ ---15 2.2.3 Pathological problems ................................................................................... 16 vii Chapter Page 2.3 Air ventilation systems for potato storage - - - - - --18 2.3.1 Types of air ventilation system - -- ..... - - _____ .18 2.3.2 Structures of ventilation system .................................................................... 19 2.3.3 Duct spacing and duct Size . - ............... - ............................. 21 2.3.4 Duct opening andits location - ................ - -- -22 2.3.5 Ventilation rate for potato storage ................................................................ 25 2.4 Equations of predicting the behaviorof fluid flow through porous media ........... 27 2.4.1 Darcy’s equation ............................................................................................. 27 2.4.2 Muskat’s equation .......................................................................................... 29 2.4.3 Shedd’s equation ............................................................................................ 33 2.4.4 Hukill’s equation ............................................................................................ 35 2.4.5 Sheldon’s equation ......................................................................................... 37 2.4.6 Bear’s eqaution ........................................................................................ 39 2.4.7 Brooker’s equation ......................................................................................... 40 2.4.8 Segerlind’s equation ....................................................................................... 41 2.4.9 Other equations ............................................................................................... 43 . 2.5 Representing fluid flow through porous media ...................................................... 45 3 ANALYSIS METHODS ................................................................................................ 68 3.1 Establishment of mathematical models ................................................................. 68 3.1.1 Models of pressure and velocity distributions .............................................. 68 3.1.2 Models of streamlines and flow rates ........................................................... 71 3.2 Application of finite element method ..................................................................... 73 viii Chapter Page 3.2.1 Applying the Galerkin method ...................................................................... 74 3.2.2 Applying the Green-Gauss theorem .............................................................. 75 3.2.3 J acobian transformation ................................................................................. 77 3.2.4 Gauss-Legendre quadrature ........................................................................... 81 3.2.5 Applying the direct stiffness method ............................................................ 82 3.2.6 Applying the Newton-Raphson method .......................................... . .............. 8 2 3.3 Compilation of computer programs ........................................................................ 85 3.3.1 Calculation of nodal pressure values ............................................................ 85 3.3.2 Calculation of pressure and velocity for the selected cross-section ............ 88 3.3.3 Calculation of streamlines and flow rate for the selected cross—section ..... 9O 4 ANALYSIS OF AIR FLOW PATTERNS ..................................................................... 93 4.1 Preparation of basic data .......................................................................................... 93 4.1.1 Basic data of duct system and potato storage ............................................... 93 4.1.2 Selection of variables and their levels ........................................................... 94 4.1.3 Coefficients in Shedd’s equation ................................................................... 96 4.1.4 Defining boundary conditions ......................................................................... 97 4. 1 .5 Calculation of duct pressures ......................................................................... 99 4.1.6 Mesh generation ............................................................................................ 102 4.2 The effect on air flow patterns of variables under study ....................................... 107 4.2.1 The effect of duct spacing on air flow patterns ............................................ 108 4.2.2 The effect of duct Size on air flew patterns .................................................. 124 4.2.3 The effect of potato pile depth on air flow patterns .................................... 136 ix Chapter Page 4.2.4 The effect of duct pressure on air flow patterns........ ........................................ 140 4.2.5 The effect of duct shape on air flow patterns ................................................... 141 4.2.6 The effect of the distance to the duct enhance on air flow patterns ............... 142 5 CONCLUSIONS AND RECOMMENDATIONS ......................................................... 144 5.1 Conclusions ............................................................................................................. 144 5.2 Recommendations ................................................................................................... 147 LIST OF REFERENCES ................................................................................................... 148 Table 1 LIST OF TABLES Total potato production, utilization, and shrinkage and loss in the United States for five selected years ............................................ The nutrient composition of potatoes ................................................... Respiration rate of potato tubers for the British cultivars Arran Consul, King Edward and Majestic .............................................. Potato bed depth vs the required length of 1.9 cm slot for in-floor rectangular duct in the Pacific Northwest ................................. Potato bed depth vs spacing of discharge hole for circular duct in the Pacific Northwest .................................................................. Selected variables and their levels as used m the computer Page .................... 6 .................... 7 .................. 11 .................. 23 .................. 23 95 P19813118 ...... - -_ __ -- Coefficients A and B of Equation [13] for air flow through potato storage .......................................................................................... The effect of duct Spacing on air flow patterns. related data and Figure numbers .................................................................................. The effect of duet Size on air flow patterns: related data and Figure numbers .................................................................................. ................... 97 ................. 111 ................. 127 LIST OF FIGURES Figure Page 1 The relationship between storage temperature and sugar contents of potato tubers (cv. Majestic) ........................................................................................ 14 2 The typical changes in sucrose concentration during growth and Storage of potato tubers ....................................................................................... 14 3 Typical arrangements of main plenum and lateral ducts for potato Storage .............................................................................................................. 20 4 Effect of equivalent diameter on the uniformity of air discharge ........................... 21 5 Effect of the ratio of discharge area to duct cross-sectional area on the uniformity of air discharge ..................................................................... 24 6 Patterns of temperature, moisture, pressure and Streamline distribution in a grain drying bin .............................................................................. 46 7 Air flow patterns in the section of an oat drying bin .............................................. 47 8 Air distribution patterns for different duct shapes and spacings .............................. 48 9 Pressure patterns established by numerical method and by experimental data .................................................................................................. 50 10 ISO-flow lines, iso-pressure lines and Streamlines for different duct openings ............................................................................................... 51 11 Velocity distribution in the lower portion of the bin for three duct pressures .................................................................................................... 53 12 Calculated pressure patterns expressed as percentage of . duct pressure ................................................................................................................ 54 13 Pressure patterns obtained from experimental data for different duct treasures" ....... - - ................ - ....................................... 55 14 Air pressure and flow path patterns for different grain repose angles .............................................................................................................. 57 xii Figure Page 15 16 17 18 19 20 21 22 23 25 26 27 28 29 30 31 Airpressureandflowpathpattemsforasectionofa conical Shaped pile ..................................................................................................... 57 Calculatediso-pressmelinesinagraindryingbinwith different duct shapes and openings ............................................................................ 59 Pressunecontourlinesintheenn'anceregion .......................................................... 61 Pressure distribution for Y-shaped duct ................................................................... 62 Velocity distribution for Y-Shaped duct ................................................................... 63 Pressure contour lines for grain bed with different duct pressures ..................................................................................................................... 65 Velocity vector fields for bins with semicircular and rectangular ducts ........................................................................................................ 66 Air flow patterns for flat grain Storage with three circular ducts .............................................................................................................. 68 Sketch of twenty-node solid element with natural coordinates ................................................................................................................. 78 The flow chart of the computer program for the calculation of nodal pressure values in 3-D space ...................................................................... 87 The flow chart of the computer program for the calculation of pressure and velocity in 2-D Space ...................................................................... 89 The flow chart of the computer program for the calculation of streamline and flow rate in 2-D space ................................................................... 92 Sketch of boundary conditions for potato storage .................................................. 98 Friction coefficient as a function of Reynolds number for round pipes of various relative roughness ratio es/d ............................................... 101 Mesh generation for potato storage with triangular duct ........................................ 103 Mesh generation for potato storage with circular duct ........................................... 104 Mesh generation for potato storage with semicircular duct .................................... 105 xiii Figure Page 32 33 34 35 36 37 38 39 40 41 42 43 45 Mesh generation for potato storage with rectangular duct ..................................... 106 ISO-pressure lines and streamlines (left) and velocity profiles (right) fortriangularductwithductspacingof 1.8m ......................................................... 112 ISO-pressure lines and streamlines (left) and velocity profiles (right) fortriangularductwithductspacingof 2.4m ......................................................... 113 ISO-pressure lines and streamlines (left) and velocity profiles (right) fortriangularductwithductspacingof 3.1 m ......................................................... 114 ISO-pressure lines and streamlines (left) and velocity profiles (right) for circularductwith duct spacing of 1.8 m ............................................................ 115 ISO-pressure lines and streamlines (left) and velocity profiles (right) forcircularduct with duct spacing of 2.4 m ............................................................ 116 ISO-pressure lines and Streamlines (left) and velocity profiles (right) for circular duct with duct spacing of 3.1 m ............................................................ 117 ISO-pressure lines and Streamlines (left) and velocity profiles (right) for semicircularduct with duct spacingof 1.8m ..................................................... 118 ISO-pressure lines and streamlines (left) and velocity profiles (right) for semicircular duct with duct spacing of 2.4 m ..................................................... 1 19 ISO-pressure lines and Streamlines (left) and velocity profiles (right) for semicircular duct with duct spacing of 3.1 m ..................................................... 120 ISO-pressure lines and streamlines (left) and velocity profiles (right) for rectangular duct with duct spacing of 1.8 m ...................................................... 121 ISO-pressure lines and streamlines (left) and velocity profiles (right) for rectangular duct with duct Spacing of 2.4 m ...................................................... 122 ISO-pressure lines and Streamlines (left) and velocity profiles (right) for rectangular duct with duct spacing of 3.1 m ...................................................... 123 ISO-pressure lines and streamlines (left) and velocity profiles (right) for triangular duct with duct Size of h, x a, = 0.64 m x 0.36 m ............................. 128 ISO-pressure lines and streamlines (left) and velocity profiles (right) for triangular duct with duct Size of h, x a, = 0.67 m x 0.39 m ............................. 129 xiv Figure Page 47 ISO-pressure lines and streamlines (left) velocity profiles (right) forcircularductwithductsizeofdc=054m ........................................................ 130 48 ISO—pressure lines and streamlines (left) velocity profiles (right) forcircularductwithductsizeofdc=058m ........................................................ 131 49 ISO-pressure lines and streamlines (left) velocity profiles (right) for semicircular duct with duct Size of r, = 0.38 m ................................................. 132 50 ISO-pressure lines and Streamlines (left) velocity profiles (right) for semicircular duct with duct size of r, = 0.41 m ................................................. 133 51 ISO-pressure lines and streamlines (left) velocity profiles (right) forrectangularductwitlrductsizeofa,xb,=0.34mx0.34m ........................... 134 52 ISO-pressure lines and streamlines (left) velocity profiles (right) forrectangularductwitlrductsizeofa,xb,=0.36mx0.36m ........................... 135 53 ISO-pressure lines and streamlines (left) velocity profiles (right) for triangular duct with depth of potato pile of 3. l m ............................................. 138 54 ISO-pressure lines and streamlines (left) velocity profiles (right) for triangular duct with depth of potato pile of 5.5 m ............................................. 139 LIST OF SYMBOLS Dimensions: Lalength, M=Mass, t=Time, and T=Temperature [B] [B]T b sq C C C Constant in Equation [3]. [4]. [5]. [7]. [10]. [12]. [13]. [15]. [17]. [18]. [27] Cross-sectional area in Equation [1] (L’) Flow cross-sectional area in Equation [73] (L7) late-21 duct alluring area (I!) Cross-sectional area Of lateral duct (L7) Cross-sectional area of main plenum (L1) Jacobian matrix in Equation [67], [68] Half of the width of rectangular duct (L) Half of the base length of triangular duct (L) Notation of adjoint matrix in Equation [59] Comm“ in Equation [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. [12]. [13]. [15]. [17]. [18]. [25]. [27]. [30] Row vector of the first order derivatives of [N] in Equation [52], [53], [58], [61], [63] Transposed [B] in Equation [52], [61], [63] Height of rectangular duct (L) Con-mm in Equation [3]. [8]. [10]. [12]. [28]. [29] State of flow factor in Equation [6] Potato storage capacity (M) cosa Direction cosine in Equation [50] xvi cosB cosy cosh Direction codne in Equation [50] Direction cosine in Equation [50] Hyperbolic cosine in Equation [16]‘ Equivalent diameter in Equation [18] (L) Hydraulic diameter in Equation [73] (L) Diagonal matrix in Equation [52] Diameter of duct in Equation [70], [71] (L) Effective diameter of granular particle in Equation [4], [6] (L) Notation of differential in Equation [38], [39], [42], [43], [47], [50], [52], [60], [61] Diameter of circular duct (L) Notation of determinant in Equation [59], [60], [61], [63] Notation of divergence Notation of element in Equation [47], [49], [51], [52], [61], [62] Exponent in Equation [30] Function of natural coordinates in Equation [65], [66], [67], [68] Force vector in' Figure 24 Friction factor in Equation [70], ['72] Friction factor in Equation [18] Leva’s friction factor in Equation [6] percentage distribution of fine materials in Equation [12], [17] Particle shape factor in Equation [6] Acceleration of gravity in Equation [4], [6], [69], [70] (Lt’) xvii [1]“I K K K Specific gravity of potato ML") Height of packed bed in Equation [1] (L) Depth of slain bed in Equation [18]. [28] (L) Height of potato storage (L) Depth of potato pile (L) Hydraulic head in Equation [1] (L) Hydraulic head in Equation [1] (L) Friction loss in Equation [69], [70] (L) Height of triangular duct (L) Jacobian matrix in Equation [56], [57], [59], [60], [61], [63] Inverse matrix of [r] in Equation [5 8], [59] Constant in Equation [28], [29], [31] Hydraulic conductivity in Equation [1] (Lr‘) permeability in Equation [21] [KM] Element stiffness matrix in Equation [51], [52], [61], [62] [K] Kn itG N < f ”53‘ Global stiffness matrix in Equation [64] Modified Ergun product constant in Equation [7] Granular permeability in Equation [26], [27], [36], [37], [47] Function of velocity in Equation [17] Permeability related to local coordinate in Equation [20] Permeability related to local coordinate in Equation [20] Permeability related to local coordinate in Equation [20] Permeability of porous media in Equation [2] (L’) xviii 900“ F 50 W -° Rte) Length of potato storage (L) Duct Spacing (L) Natural logarithm in Equation [15], [17], [19] Common logarithm in Equation [14] Function of pump work in Equation [69] (L) Coefficient in Equation [25] Moisture content in Equation [19] Number of lateral duct Row vector of shape function Transposed [N] in Equation [47] Notation of normal direction Pressure (ML‘t'z) Lateral duct pressure (ML“t") Wetted perimeter in Equation [73] (L) Dimensionless pressure parameter in Equation [14] Volumetric flow rate in Equation [1] (L’t‘) Function of Cartesian coordinates in Equation [50] Air flow rate in lateral duct (L’t") Air flow rate per unit mass (L3t‘M“) Function of Cartesian coordinates in Equation [50] Ratio of open area of duct system to floor area served by duct system Source strength in Equation [16] Contribution of element to the residual equation in Equation [47] xix Rea rt S Reynolds number in Equation [71], [72] Radius of semicircular duct (L) Curved surface in Equation [50] sinh Hyperbolic sine in Equation [16] < Time Function of Cartesian coordinates in Equation [50] Upper triangular matrix in Figure 24 Velocity vector Velocity 01‘) Volume in Equation [47], [52], [60] (L3) Air flow velocity exiting lateral duct (Lt‘) Air flow velocity in the lateral duct (Lf') Mean fluid velocity over duct cross-section in Equation [71], (Lt') Velocity in normal direction (Lt‘) Air flow velocity in main plenum (Lt‘) Velocity component in Cartesian coordinates (Lt‘) Velocity component in Cartesian coordinates (Lt‘) Velocity component in Cartesian coordinates (Lt') Width of bin in Equation [16] (L) Width of potato storage (L) Water gauge (L) Weighting coefficients in Equation [62] Cartesian coordinates (L) Yl 3’2 Cartesian coordinates (L) Cartesian coordinates (L) Cartesian coordinates (L) Elevation head in Equation [69] (L) Elevation head in Equation [69] (L) Cartesian coordinates (L) Cartesian coordinates (L) Porosity of porous media in Equation [4], [6] Preset comparative value in Figure 24 Roughness in Equation [70] and Figure 28 (L) Natural coordinate Natural coordinate Dynamic viscosity of fluid in Equation [2], [4], [5], [71] (ML"t") Natural coordinate Natural coordinates of sampling points The ratio of the circumference of a circle to its diameter in Equation [16] The r numbers in Equation [11] Density Of fluid in Equation [4], [5], [6], [32], [69], [71] (ML”) Dry matter bulk density in Equation [19] (ML”) Summation in Equation [62] Stream function in Equation [41], [42], [43], [45], [46] Notation of partial derivative CHAPTER 1 INTRODUCTION 1.1 Background Potatoesareanimportantagriculturalcropandareamainfoodinthedailydietof the United States. In 1987 potato production reached 17.7 million metric tons, among which 50.3%, 33.5% and 8.1% were used for processing, for table stock and for other usages (such as seed and feed), respectively. However 8.2% were lost due to damage, shrinkage, deterioration or other factors (USDA Agricultural Statistics, 1989). In a modern potato production system, potatoes will be put in storage immediately after harvest to reduce losses and to preserve quality for later marketing or processing. During storage, the quality of potato tubers may deteriorate due to physical, physiological and pathological problems. The extent of deterioration, among other things, is closely related to the temperature, relative humidity and composition of the air within the potato storage. To maintain a suitable temperature, relative humidity and air composition, forced air ventilation systems are commonly used in commercial potato storages. A uniform air flow through the potatoes in storage is desirable to achieve ideal air conditions and consequently to help maintain the quality of the stored potato tubers. Therefore, the investigation of pressure and velocity distributions within stored potatoes becomes an 1 important research subject. Fluid flow throughporous media, suchasairflowthroughapotatopile, isasimple physical process. Yet, numerous researchers in various fields have analyzed the factors that affect this process and have attempted to formulate mathematical prediction models. This fact not only indicates the importance of the subject of fluid flow through porous media in the relevant engineering areas, but also implies the difficulty in describing the phenomena of fluid flow through porous media. Air flowing through a potato pile will encounter resistance from potato tubers or dirt, causingapressuredroptodevelop. Airflowpatternsinapotatostoragearestrongly affected not only by characteristics of potato itself such as tuber size, porosity, orientation and cleanness, but also by air properties and flow parameters, and by the configurations of the air duct system and the storage. Because of the inhomogeneous distribution of potato size and porosity, and the nonuniform distribution of air flow along the duct length, air flow phenomena in a potato pile will show nonuniformity both in two- and three-dimensional spaces. By taking into consideration these factors and their mutual relationships, mathematical models based on physical principles and experimental data can be built. These models, which govern the relationship between pressure gradient and air velocity, will provide a useful means for studying air flow patterns in a potato pile and for evaluating the effect of various factors on these patterns. Since these models involve nonlinear partial differential equations, they can not be solved analytically (Ames, 1965). The finite element method, which has been proved to be an effective numerical method for solving field problems (Segerlind, 1984), was used in this study to facilitate the solution process. The pressure, velocity and air flow 3 direcfionatanypomtwithinthepilemtwo-andmreedimennonaldomainscanbe calcuhtedandfieairflowpahernsindwfomsofiscpressumfinesmfieamfinesand velocity profiles can be plotted. 1.2 Objectives Thegeneralpmposeofthisstudywastoanalyzeairflowpatterns withinapotatopile in two- and three-dimensional spaces. In order to reach this goal, mathematical models of the process were developed. Using the finite element method, computer programs were written to calculate the specific solutions for the mathematical models. The specific objectives for analyzing the calculated results were: 1. To present graphically the iso-pressure lines, streamlines and velocity profiles; 2. To predict the common tendency of air flow patterns in a potato pile; 3. To analyze the effect on air flow patterns of duct spacing, duct size, duct pressure, duct shape, depth of potato pile and the distance from a selected cross-section to the duct entrance. CHAPTERZ UTERATUREREVIEW 2.1 The potato crop Thepotatoisanancientdomestieatedcrop. ItoriginatesfromtheAndesofPeruand BohviainSouthAmerica.Tlfiscropwasfintinu'oducedinmtheNormmgican continent from England via Bermuda in 1621 (Hawks, 1978). TherearetwomaincroppingseasonsintheUnitedStates: early-croppotatoesand late-crop potatoes. The early-crop potatoes are mostly planted in the Southern and Western states and harvested during the spring and summa months. The late-crop pommesuephnwdmmeNoMemhalfofthiscounuymdarehawesteddudngfltelate summer and fall months. Most of the late-crop potatoes are stored, with about half of the stored potatoes used for processing purposes (Hardenburg et al. 1986). The total production for each of five selected years is shown in Table 1 (USDA Agricultural Statistics, 1989). The utilization of the potato has changed, with potatoes processed for food increasing significantly. The processed potato, including potato chips, frozen french fries, dehydration, etc., accounted for only 10.1 percent of the total production and 13.7 percent of the total use for food in 1956, but it reached 50.3 percent and 60.0 percent 4 5 in 1987, respectively. In contrast, potatoes used for fresh food has declined markedly as shown in Table l. The potato is rich in nutrients. It provides significant quantities of food energy, protein and vitamin C. Table 2 gives the nutritional values in percentages of the U.S. Recommended Daily Allowances established in 1973 (revised in 1980). It is obvious that potato can provide an important source of vitamin C in the daily diet. To provide a common language for commerce and to set a minimum quality level, the United States Department of Agriculture developed the Potato Grade Standards. These standards specify U.S. extra #1, U.S. #1, U.S. commercial and U.S. #2 grades. Among these grades U.S. #1 is the principal trading grade. Visual inspection is used most often to evaluate the quality of potatoes. Potatoes of any kind and size should have a relatively smooth, clean and well shaped appearance without badly cut and bruised skin and without any green part from light exposure. They also should have a firm texture without wilt and sprouting (Seelig, 1972). Besides the above visual qualities, stored potatoes for processing purpose should be mature, not stressed, free from imperfections, low in reducing sugars (less than 0.25 percent) and high in specific gravity (Gould, 1984). Table 1. Total potato production, utilization, and shrinkage and loss in the United States for five selected years. Source: USDA Agricultural Statistics, 1989 Year 1960 (Unit: 1,000 metric ton) 1956 1970 1980 1987 Total production 11,159 11,673 14,782 13,797 17,675 Total for fresh food 7,054 7,006 5,950 4,516 5,912 Total for processed food 1,123 2,224 5,811 6,857 8,883 Total shrinkage and loss 696 589 1,088 1,055 1,448 Total for other usages 2,286 1,854 1,933 1,369 1,432 Fresh food to 63.2 s 60.0 95 40.3 95 32.7 96 33.5 96 total production Fresh food to total potato ‘ 86.3 as 75.9 as 50.6 75 39.7 % 40.0 75 used for food Processed to total production 10.1 % 19.1 96 39.3 96 49.7 96 50.3 % Processed to total potato 13.7 96 24.1 96 49.4 96 60.3 96 60.0 96 used for food Shrinkage and loss 6.2 % 5.1 96 7.4 96 7.7 96 8.2 % to total production Other usages to 20.5 96 15.9 96 13.1 % 9.9 % 8.1 % total Production Table 2. The nutrient composition of potatoes Source: Thornton and Sieczka, 1980 Nutrient Nutrient % U.S. RDA in 96 U.S. RDA in Values medium potato large potato (about 150g) (about 250g) About 110, approx. About 180, approx. Calories 4 96 of total calories 7% of total ealories , for adult male for adult male Vitamin c 13.2 - 54.2 mg 56.6 % 93.3 % Iodine 0.05 - 0.04 mg 15.2 96 25.3 96 Vitamin a, 0.20 - 0.60 mg 16.4 76 27.3 % Niacin 1.00 - 3.50 mg 12.1 96 20.2 96 Copper 0.10 - 0.50 mg 16.9 96 28.2 96 Magnesium 0.03 - 0.04 g 7.8 95 13.0 96 I Thiamin(B1) 0.07 - 0.15 mg 8.7 % 14.5 % Phosphorus 0.05 - 0.10 g 7.3 96 12.2 % Protein 2.50 - 3.60 g 4.7 95 7.8 75 H Folic Acid 7.80 - 32.5 mg 4.9 75 8.2 96 Iron 0.40 - 2.10 mg 5.2 % 8.7 96 Riboflavin(B2) 0.03 - 0.10 mg 3.6 96 6.0 96 I Zinc 0.50 - 04-80 mg 3.9 96 6.5 96 8 2.2 Causes and control of deterioration of stored potatoes 2.2.1 Elysical problems Potatoes that are mechanically harvested may be injured by blades, chains and other moving parts. The injured potato tubers, if left unprotected, will have high evaporation losses and willbeeasily affectedbydiseases. Therateandextentofthewoundhealing process are mainly affected by temperature, humidity and composition of the air surrounding the tubers. Wilson (1967) suggested that, immediately after being placed in storage, potatoes be cured by holding at a temperature of about 10° to 15.6" C and a relative humidity of above 90 percent for 10 to 14 days to permit suberization and wound periderm formation. Earl (1976) proposed a higher temperature of 15.6° to 21.1" C and a higher relative humidity of 95 percent. Although wound periderm formation is most rapid at about 21° C, Hardenburg et al. (1986) maintained that a temperature of 10" to 15.6" C would be preferable, as decay to the injured part is more likely to occur at a higher temperature. While temperature plays an important role in the process of suberization, the relative humidity level has more influence than temperature on a desirable new dense periderm. Potatoes stored in an environment with a high relative humidity of 95 percent will suberize at temperatures of 7.2° to 18.3° C (Cargill, 1976). Hammerschmidt and Cameron (1986 and 1987) investigated the effect of co2 level on wound healing. They found that as the C02 level increased from 0.0% to 8.0% the rate of wound healing gradually declined and the degrees of the soft rot decay gradually 9 increased. Therefore, increases in CO, duringthe early phase of storage can have a very negativeeffectonthepotatotubers. Dudngdreloadingandudoadingprocessesinthepommuomgemommesmayalso sufferinjury.Cargill(1976)recommeodedthatthedropheightbelimitedto30to40cm andthatconveyorspeedsnotexceed40m/min.Cargill(l976)andRastovski etal. (1987) also noted that during storage,pressmebruisewillposeaserious problemtothe lower layer of potatoes, since the deformed or damaged tissue is particularly sensitive to blue-grey discoloration. To help reduce pressure bruise, Cargill (1976) recommended thatthedepthofthepotatostoragebelimitedto3.7to4.6m, andtherelativehumidity in thestorage shouldbekeptat92to95percent. 2.2.2 lllysiological problems 2.2.2.1 Water loss The potato tuber contains 74 to 82 percent water (Burton, 1989). Water loss will directly affect both the weight loss and the appearance of the potato. Shrinkage and loss account for 5 to 8 percent of the total potato production as presented previously in Table l. Cargill (1976) observed that if potato weight loss reaches 5 percent of the original weight, the potato will shrink; if this loss reaches 10 percent of the original weight, the potato will become wrinkled and spongy, difficult to peel and virtually unsalable. Water loss from stored potato tubers is mainly through evaporation and respiration. It is closely related to the temperature and humidity of the air in the storage, in addition 10 to factors such as cultivar, tuber size, maturity and injury condition. Under constant humidity conditions, higher temperatures result in a higher vapor pressure deficit and higher water loss. Under constant temperature conditions, lower humidifies result in a highervaporpressure deficitandhigherwaterloss. Therefore, potatoweightlosscanbe mkfimizedbyreduchgtheaomgetempuammtomducevaporpressumdefidtmdmber respiration rate, and by increasing the storage relative humidity to reduce vapor pressure deficit and moisture exchange. Hardenburg et al. (1986) recommended that to minimize weight loss the optimum temperature range for storing most cultivars of potatoes to be processed intochipsorfrenchfriesbebetween 10°and 13°Candthedesirablerelative humidity be 95 percent. Rastovski et al. (1987) proposed a lower temperature for long term storage: 7° to 10° C for chipping potatoes and 5° to 8" C for french frying potatoes. 2.2.2.2 Respiration The potato is a living organism, and respiration is the metabolic process necessary for maintaining the life of the potato tuber. During this process the sugars in the tuber are converted into carbon dioxide, water and heat energy through the consumption of oxygen. The whole process can be described by the following relationship: C,H,,0, + 602 at 6C0, + 611,0 + Energy A Under standard conditions, for example, the oxidation of 180 g (1 mole) of glucose with 192 g (6 mole) of oxygen will release 264 g (6 mole) of carbon dioxide, produce 108 g (6 mole) of water and yield 2,880 k] of energy among which about 30% is fixed in ATP (metabolic energy) and 68 % is released into surrounding media in the form of heat 11 (Stryer, 1975 and Rastovski et a1. 1987). Obviously, respiration will directly result in thelossofdrymatterfromthepotatotuber.Butthemainproblemsrelatedtorespiration ofthestoredpotamesamtemperatumhwreases,flleaccumulafimofC0,andflie depletionofoz. Asrespirationrateincreases, moreheatwillbereleased,whichwill increasedeeayandsenescenceofthestoredpotatoes.Ahighaccumulationofcozand the depletion of Ozareharmfultothepotato tubers (Burton, 1989). Respiration rate is strongly affected by temperature. The classical concept of the effect of temperature is that the respiration rate for biological materials will generally double forevery10°Cfiseintemperamre.Thiswndencycanbeseenveryc1earlyinTable3, in which Burton (1989) cited the data of respiration rates for a batch of healthy and mature potatoes (cv. Arran Consul, King Edward and Majestic) one month after harvest. Table 3. Respiration rate of potato tubers for the British cultivars Arran Consul, King Edward and Majestic. Source: Burton, 1989 Temperature Release C02 Absorption 02 Heat generation °C 10’mg/kg s 10’mglkg s 10’J/kg s ' 0 2.64 1.92 5 1.27 0.92 12 ItwasnotedthattherespirafionrateisveryhighatthetemperatureofO'C.Burton (1989) implied that in this case the high rate ofrespiration at temperatures below 5" C may be related to sucrose accumulation at the low temperatures. Hunter (1985) observed the effect on the respiration rate of difi'erent storage temperatures (3.3° C, 7.2° C and 10.0° C). He concluded that the direct effect of temperature on respiration rate in potatoes is of relatively short duration (7 - 10 days). Generally, thedeclineandincreaseoftherespirationratearemorerapidathigher storage temperatures during the falling rate period and the rising rate period after the end of dormancy, respectively. For a long storage period, the respiration is often minimized at7.2°C. Hefunhernoteddrattheweightlossmteiscloselyrelatedtodlerespimfion rate, especially at high relative humidity and low vapor pressure deficit. Both respiration rate and weightlossratecanbeexpressed bythesametypeofexponential functionwith different coefficients:A[eltp(-kt)]+C for the falling rate period, and A[exp(kt)-1]+C for the rising rate period. Theevaporationofwaterinan unventilatedstockofpotatowillremoveabouthalfthe metabolic heat production. Adequate ventilation to remove the heat buildup due to respiration and to maintain suitable levels of CO, and 0, is still very important. 13 2.2.2.3 Sugar level If potatoes are stored at a low temperature for a long time, a biological transformation will occur in the tuber. The starch will be gradually transformed into reducing sugars (glucose and fructose) and nonreducing sugar (sucrose). Burton (1982) observed the sugar contents of potato tubers (cv. Majestic) after 4 weeks of storage (17th Dec. to 14th Ian.) at various temperatures. He noted that sugar content increases very markedly when the temperature is below 10° C, as shown in Figure 1. Cash et al. (1986 and 1987) also noted that the sucrose, glucose and fructose contents of the Russet Burbanks and Atlantic potatoes increased during storage. The color of potato chips made from these potatoes became darker as the storage time increased. Sowokinos and Preston (1988) developed the method of Chemical Maturity Monitoring (CMM) to analyze the sucrose and glucose contents within potato tubers during their grth period and storage period. Typieal changes in sucrose concentration are shown in Figure 2. It is very clear that after several months of storage the sucrose level in potato tubers will gradually increase, especially during the senescent sweetening process. They suggested that for processing potatoes the maximum tolerable concentration levels of sucrose and glucose are a Sucrose Rating (SR) less than 1.0 (mg sucrose/g fresh tuber) and a glucose level less than 0.035 (mg glucose/g fresh tuber). They also thought that the ventilation stress after harvest may cause the sucrose values to increase to an SR of 2.0 or above. Therefore, storage management can be improved by monitoring the sucrose and glucose levels of the potato tubers. 14 Sugar(% trash wt) 3‘ 0 Total sugar 21 D Glucose + Fructose A Sucrose 1 0 T r r I r r T I O 2 4 6 8 10 12 14 16 Temperature °C Figure 1. The relationship between storage temperature and sugar contents of potato tubers (cv. Majestic). Source: Burton, 1982. '° «cm—um» 0—DORHANCY—U ‘ k 9 far S‘IURAGE - KPCJO'F a g 8- x L’s" W‘W—fi'mevmi 1|» ’; 7 - 'REVERSIBLE' . \ g 6" mcSTAfCH ‘ § 5- " slcaoss J' a 1 senescsm ” swearsums ? x ‘ E 3. Remus lama: / g m X 2' ENERGY ,/ ‘ | l- \X XX. ’X’ . L 2 kg.‘ H I 3‘ rx’ 0 a a 1 L 1 a JUL “SEPT.” WV. *6. MN. FEWAPRILIAY JLNE Figure 2. The typical changes in sucrose concentration during growth and storage of potato tubers. Source: Sowokinos and Preston, 1988 15 Cargill (1976) and Earl (1976) concluded that a high content of reducing sugars is undesimbleinpotamesdesfinedforpommchip,fiawhffiesandoflrerdehydmted products,asthehighwncenuafimofmgamudnrendtinduk-coloredprowssed products.Theysuggeueddutthestongetemperamresbebetweul7°and10°Cw prevent the transformation of starch to sugar. 2.2.2.4 Sprouting Sprouting of stored potatoes should be prevented. Sprouting will increase water loss. Rastovski et al. (1987) noted that moisture loss through theepidermis of thepotato sproutsisabout 100to 150timesas muchasthatthroughtheintactperiderm ofamature tuber. As the result of sprouting, potato tubers will shrivel and lose their market values. Thespmufingofmbersinstoragewinalsoincreasetheresistancetoairflow, thus increasing the pressure head losses. Seelig (1972) stated that potatoes will not sprout until two to three months after harvest, even at temperatures of 10° to 15° C. But after two to three months, when the storage temperature rises above 4.4° C or when the temperature fluctuates, the dormancy of potatoes will be broken and sprouting will occur. Thedormancyperiod nrainlydependsonthecultivarofthepotato. Butasfaras storage environment is concerned, maintenance of lower and stable temperature is preferable to help prevent sprouting. Cargill (1976) thought that the sprouting was minimized at temperatures below 4.4° C and almost nonexistent at temperatures around 2.2° C. Wilson (1967) reported that if sprout inhibitors are used the storage temperature 16 maybekeptashighas7.2°C;butwithoutsproutinhibitorsatemperatureof4.4°Cwill benecessarytopreventsprouting.Hesuggestedthattheactual storagetemperaturebe acompmnfisebetweenthetemperahuempmventsproufingandflreonewremmme conversionofstarchtosugar. Therefore,thephflosophyofapotatostorageistoretainwamrinthepotatotuber, keeptherespirationratetoaminimum,holdthereducing sugarstoalowleveland maintain the external appearance of the stored potatoes (Plissey, 1976). 2.2.3 Pathological problems Hide and Lapwood (1978) reported that the potato is prone to more than one hundred diseases during its whole living period. Diseases that cause the deterioration of the stored potatoes are fungal diseases (such as late blight and silver scurf), bacterial diseases (such as bacterial soft rot, brown rot and ring rot), and storage pests (such as potato tuber moth and fruit flies). Diseases require a suitable environment to survive and to develop. Among other factors, temperature and relative humidity in the storage are the most important factors that can be used to curb diseases. For instance, Rastovski et al. (1987) indicated that the prime requirement to control potato blight is for the tuber surface to be dry. Storage under warm dry condition for a few weeks is sufficient to help control blight. A storage temperature below 3° C and a relative humidity below 90 percent will prevent the spread of silver scurf. While the potato tuber moth is a dread parasite in tropical countries, it is well controlled at storage temperature below 10° C. 17 Campbell (1962) and Cargill (1976) also noticed that free water dripping from the ceilingontopotatoesorconderrsationofwateronthecoolerpotatoesinthepilewill cause wet potato tubers which may result in bacterial soft rot infection and subsequent wet breakdown. They emphasized that every precaution should be talnen to eliminate free water on potatoes in storage. 18 2.3 Airventilationsystarnsforpotatostorage 2.3.1 Types of air ventilation system Therearetwowaystoaerateapotatostorageznammlventilafionandforced ventilation. Natural ventilation by free convection is very slow, inefficient and uneven. Itismostoftenusedinsmallstorages. Inmodernpotatostorages, forcedventilationis a common practice. It has the advantage of controlling the storage condition rapidly, easily and accurately. Wilson (1976) and Cloud (1976) thought that the forced ventilation system should blow theairupthrough thepile ofpotatoes while theairismaintained at thepropertemperatureandrelativehumidity. Theybelievedthatthistypeofverrtilation system will give faster and more uniform cooling of the potatoes than a ”shell" ventilation system where the air is moved around and above the stored potatoes. Hunter and Yaeger (1972) proposed using a cross flow circulation system. But this type of ventilation system was not effective in maintaining uniform temperature and humidity. Also the storage width would be limited to about 3.0 to 5.0 m for efficiently controlling the air flow. This system has not been adopted commercially. 19 2.3.2 Structures of ventilation system In order to ventilate a potato storage uniformly, duct systems are widely used as the air distribution system. Typical arrangements of the main plenum and lateral ducts are shown in Figure 3. The lateral ducts are placed either in—floor or on-floor. The in-floor duct system is permanalt and is favored for ease of bin loading and unloading. The cross-section of this duct system is usually rectangular and the installation investment will be higher as compared with that of an on-floor duct system. The on-floor duct system is easy to place but it is not convenient for loading and unloading. The original investment for on-floor duct systems may be low, but the costs of repair and replacement of ducts will be significant. The cross-sections of this duct system may be triangular, circular, semicircular, or rectangular. Main Duck quarter points Main Fans at l<><>l l l°<> A Lateral Ducts) Main Fan . in the center Mann Du?) 1<><=I C J J J up E X Lateral Ducts’ n Duct Lateral D Main Fan at one end Main Duct , I , 8 Lateral Ducts Lateral Ducts v H H H M Main Duct JHUUUU j<><> TML _ Lateral Ducts ”WM 1°C J<><>l Figure 3. Typical arrangements of main plenum and lateral ducts for potato storage. Source: Cargill, 1976 21 2.3.3 Duct spacing and duct size Cargill (1976) recommended that the spacing of the lateral ducts in a potato storage be 1.8 to 2.4 or between the centers of two adjacent ducts for in-floor ducts and 2.4 m for on-floor. ducts. Cloud and Morey (1980) analyzed the effect of equivalent duct diameter (four times cross-section area divided by perimeter) on the uniformity of air discharge. Under the conditions that the duct entrance velocity was 305 m/min, duct length was 24.4 m and the slotdischargeareawasequaltotheductcross—sectional area,theyshowedthatductswith equivalent diameter of 0.24 ill have a relatively uniform air discharge along the duct length, and the air discharge near the duct entrance will increase as the equivalent diameter decreases (Figure 4). '2° I l l l l < Eotluvolcnt| Dust Diameter - 0.15m . I :' l 3 ”O r - jr‘V Equivalent Duct Dtomclcr - 0.24m ~% 0 = ‘l\ I l 2 N ' f anh- ‘-""'l 3 IOOb—‘ ‘ ~~ ' __ -___,— a“; s. i ‘ \ —v -4?’ - u U 1 Eduwolent Duct Dtometrw ' 0.61 m 3 9° ‘ / I l l l 4 l . 5 /1 Putt Characteristics . I g 80 r__l_Coostom Cross - Sutton ' _______ ' 5 Dust 61mm vctcstry- 303:.sz 2. unicorn Star Arse Otsmoutron 6 5m Ooscuoros Ano- Dust E 1W - 09mm (Moor Concrete) («er00on an. ‘ a Ova-41m Loss Cost cl Stern t 34 | 7 0'“ “n"? ' 2“"? 7 1 4 a l L L L L l ' 00 IO 20 30 4O 5O 60 70 80 90 lm DUCT ENTRANCE alsuucr Alma rue oucr. 7. Length ”as“ Figure 4. Effect of equivalent diameter on the uniformity of air discharge. Source: Cloud and Morey, 1980 2.3.4 Duct opening and its location Wilson (1976) gave the lengths of 1.9 cm slot for in-floor rectangular duct correspondingtodifferentpotatobeddepthsaslistedin'l‘able4. Thesedatawere calculated for a duct cross-sectional area of 0.25 m’, a duct spacing of 3.1 m on center, anairflow of0.5 m’lminpermetrictonofpotatoes, andaslotspacingof30.5cmalong the duct. The air discharge holes for a circular duct should be on each side near the floor and 90 degree apart. For a duct spacing of 2.4 m on center, air discharge holes with diameters of2.5, 3.2 and 3.8 cm shouldbespaced according to thedepthofthepotatoes as shown in Table 5 (Wilson, 1976). Cargill (1986) suggested that the effective slot area to the cross-section area of the lateral duct be 0.75 to 1.0, where the effective slot area is based on an air flow velocity of 305 m/min. For an in-floor rectangular duct, the potato will cover about 65 to 75 percent of the actual slot area. Therefore, the ratio of the actual slot area to the effective slot area should be 3.0 to 4.0. 23 Table4. Potatobeddepthvstherequiredlengthof 1.9 cm slot for in-floor rectangular duct in the Pacific Northwest. Source: Wilson, 1976. Potato bed depth, meter Length of 1.9 cm slot, cm Table 5. Potato bed depth vs spacing of discharge hole for circular duct in the Pacific Northwest. Source: Wilson, 1976. Potato bed Spacing of 2.5 cm Spacing of 3.2 cm Spacing of 3.8 cm depth, m diameter hole, cm diameter hole, cm diameter hole, cm 3 l 22.9 35.9 51.6 I 3.7 19.1 29.9 _ 43.2 I 4.3 16.5 25.7 36.8 4.9 14.0 22.5 32.4 I 5.5 12.7 20.0 28.9 6.1 11.4 17.8 26.0 24 Cloud and Morey (1980) also studied the effect of the ratio of discharge area to duct cross-sectional area on the uniformity of air discharge. Under the conditions that the duct entrance velocity was 305 m/min, duct length was 24.4 m and the equivalent duct diameter was 0.61 m, they found that an effective way to improve air discharge uniformity is to reduce the effective duct discharge area. This trend can be seen from Figure 5. But they noted that decreasing the effective duct discharge area would be at the expense of increased duct static pressure requirements. "° 1 l 1 l 140 r-Nunoers on Curves are the Rams at Ettecme Star / g '30 ’“DIsCharae Area to Dust Crass- / 8 Sectoanal Area Z " IZO r 5 2V / 1’" .\' 100 1- . !- O.5 1!: l 3 90 9757/ 4 Duct Characterrsucsr g .— l .0‘ Z LCWOM Cross - Search g 80 r- 2 who.” SDI Ana Damian-00 8 / 3 w - 09M (Average Camera) 8 7° >-l.5‘ «we Loss Cesium dSM-IM .— ” / 5 Dust Enron‘s veracity . 3051‘!“ 5 GO ,. / 61mm Master or bust . 0.61m __ ‘ 2.04? " °"." ""°'." ' 3‘ ‘1" I I so ‘_ 1 a #g a J 1 1 1 J 0 IO 20 30 40 50 60 70 BO 90 IOO “93%;“ orsrmc: m m: oucr. v. Length , 033*- Figure 5. Effect of the ratio of discharge area to duct cross-sectional area on the uniformity of air discharge. Source: Cloud~and Morey, 1980 25 2.3.5 Ventilation rate for potato storage Wilson (1967) proposed that, for wound healing and curing, a minimum ventilation rate of0.53 m’lmin per metric ton ofpontoes be used to remove field heat and that the air be constantly circulated during this period. For the storage period, the rate of 0.25 to 0.31 m’lmin per metric ton of potatoes will maintain the storage temperature, but the aircirculation shouldbeonanintermittentbasis. Earl (1976) recommmdedalower airflow of 0.31 m’lmin per metric ton of potatoes be used during the wound healing period. Wilkes (1976) behaved that in a normal year a minimum air flow range of 0.2 to' 0.6 m3/ min per metric ton is sufficient, but during the problem years additional air flow in the range of 0.6 to 0.9 m’lmin per metric ton should be used. Mitchell and Rogers (1976) thought that for table and seed potato storage the air flow should be 0.6 to 0.7 m3/ min per metric ton, while for chipping potato storage the air flow should be 0.9 to 1.1 m3/min per metric ton. Cargill (1976 and 1986) pointed out that a basic rule in potato storage is to use no more air than is required to maintain the storage temperature within 0.5° to 1.0“ C of the desired temperature. In a storage with tuber temperatures of 7.2° to 10° C the heat of respiration will cause temperatures to increase 0.5° to 1.0° C in 24 hours. Therefore, during the storage period the fan should not be off longer than 24 hours at one time. He noted that a minimum ventilation rate of 0.93 m’lmin per metric ton will be adequate. Forbush and Brook (1989) observed the effect of ventilation rate on the temperature, moisture and quality responses of stored potatoes. In their experiment, ventilation rates 26 upto1.9m3/minpermetrictonwereused.Theyconcludedthatweightlossand temperature control were not directly correlated to ventilation rate, and higher ventilation rates weremoreeffectiveatremoving surfacemoisnrrefrompotatoes. To effectively ventilate the potato storage, Cargill et al. (1989) and Brook (1991) suggested that the air velocity throughout the ventilation system should increase at each stage. For example, if the ratio of the total effective duct opening area to the cross- sectionalareaoftheductandtheratioofthetotalcross-sectionalareaoftheductsto the main plenum cross-sectional area are all equal to 0.75 to 1.0, then the air velocity should be230 rn/mininthemainplenum, 260 m/mininthelateralductsand305 m/minatthe outlet of the slots. These data were for potato storages in the Midwest USA. Waelti (1989) recommended a higher air flow velocities for storages in the Pacific Northwest USA. He thought the air velocity in the plenum should be 240 m/min, the air velocity at the entrance of the lateral duct should not exceed 300 m/min, and the velocity at the duct opening should be at least 381 m/min. 27 2.4 Equati-sofpredictingthebehavioroffhridflowthmnghpomusmedia The characte'istics of fluid flow through porous media have been studied extensively byreseardreminvafiousengineeingareas.1hereseamhonthissubjectisvey imporuntwdwstomgeoffarmproducm,theusageofmdegroundwater,meconuol ofreacdonsmchemiealengineeingandmcexplomfionfmpeuolerm.3ywnducflng numerous experiments on diffeent porous media under specific conditions, researches have been able to generalize equations that best express their observations. They used these equations together with other equations, such as the continuity equation, to predict fluid flow patterns in porous media. 2.4.1 Darcy’s equation The earliest study on fluid flow through porous media was conducted by Darcy in 1856 (see Muskat, 1937 and Bear, 1972). Darcy investigated the flow phenomena of water in a vertical homogeneous sand filter. The experimental results lead to Darcy’s law ‘ with the following form: . w [11 Q I-I Darcy’s law is the fundamental equation governing fluid flow through porous media, but it is limited to a very low and narrow range of Reynolds number. For liquid flow, Bear (1972) reported that it is valid for Reynolds number from 1 to 10 for practically all 28 cases, aslongasReynoldsnumberisbasedontheporediameter. Greerlsorn(l983)held thatitisonly validinthecreeping flowregimewithReynoldsnumber,basedonthe effectiveparticlediameter,lessthan 1. 'I‘hedifferenceofthevalidrangesliesinthe differert definitions of the hydraulic diameter used in calculating Reynolds number. Muskat (1937) proposed that in a general three-dimersional flow system the resultant velocity at any point is directly proportional to, in magnitude, and in the same direction as the resultant pressure gradiert at that point. So the resultant velocity may be resolved intothreecomponertvelocitiesparalleltothecoordinateaxes, eachreactingtothe pressure gradierts independently of the «has. Therefore, Darcy’s law can be expressed for an isotopic porous medium as: kaP V 8 -—-— " 11 3x kaP V I -——-— ’ u 33' v, = $3 [2] [.1 32 where k is the permeability of the media and p is the viscosity ofthe fluid. From Equation ['2] it is obvious that Darcy’s law reveals the linear relationship betweenvelocityandpressuregradient. However, thereareflowregimesthatdeviate from linearity and display non-Darcian flows. Kutilek (see Scheidegger, 1974) 29 summarized various possibilities and preseited twelve schematic flow curves for non- Darcian flows. Scheidegger (1974) discussed the physical causes of such deviation and analyzed a variety of correlation equations for nonlinear flow through porous media. He thought that the main cause of the nonlinear flow was the high flow velocity. 2.4.2 Muskat’s equation Muskat (1937) analyzed the behavior of fluid flow with high Reynolds number through porous media. He found that as the Reynolds number increases the pressure gradieitbeginstoincreasefaswrthanthevelocity. Inthiscase, thepressure gradiertwill be proportional to the square of the velocity and will be independert of the viscosity of the fluid. For viscous flow with a low Reynolds number, the pressure gradient is directly proportional to the viscosity as stated by Darcy’s law. He assumed that for the transition between viscous flow and turbulent flow the pressure gradient will be best described by the sum of terms of several powers of velocity, that will correspond to: 5‘3 . AV +BV C [3] dn whee A and a are constants, and c is intermediate between 1.0 and 2.0. Equations that have a form similar to that of Muskat’s are widely used to represent experimental data. Ergun (1952) studied fluid flow through a packed column. He consideeddratthepressurelossesarecausedbybothviscousenergylossandkinetic energy loss. Since viscous and kinetic energy" losses are the functions of the first and 30 second order of velocity, respectively, he proposed the following equation for all types of flow: 93g - ApV-t-prz [4] dn In Equation [4], A and B are constants related to the properties of the fluid, the characteristics ofthegranular solidandtheporosityofthepacked column,pisthe density of the fluid, and g is the accele'ation ofgravity. For a granular solid, constants A = lSO(l-e)’l(e’d’) and B =- 1.75(l-e)/(e’d), whee e is the porosity of the porous mediaanddisthe effectivediameterofthegranularparticle. Ifweletgbemovedto the right side of the equation and be included inside the constants A and B, then as Apv +BpV2 [5] 34% which is equivalert to Muskat’s equation. Ieva (1959) observed laminar and turbulent flows through beds packed with sphe'ical and nonsphe'ical particles. Assuming that the fluid flow will be influenced by the shape of the particle, the porosity of the media and the friction factor, he proposed that 5L: = Bfof'w2 [6] where fL is the modified friction factor, f, is the particle shape factor, B is the coefficiert related to fluid properties and porosity of the porous media: (B = 2p(l-e)’c/dge’), and 31 C is the state of flow factor that is also a function of Reynolds number. Leva’s equation canbeshowntobeaspecialcaseofMuskat’sequation. An approach similar to Ergun’s was proposed by Bakker—Arkema et al. (1969). They usedcherry pitsastheMtedmediaandmodifiedErgun’sequationwithaconstantK, tofittheirexpe'irnertaldata. Patterson(l969)determinedtheresistancetoairflowof randomly packed beds of plastic sphere, cherry pits, shelled corn and navy beans for air flow rates in the range of 3.0 to 36.0 m’lmin/m’. A modified Ergun equation, the same as Bakker-Arkema’s, was used for predicting air flow parameters. He reported that the modified equation fit the expe'imertal data well for shelled corn and navy bean. Patterson et al. (1971) further simplified Ergun’s equation for stored granular materials as follows: a; - 1|,(twtisv’) [7] where A and B are constants. This equation is similar to that of Muskat’s. Matthies and Peterson (1974) used several modified equations to relate pressure drop to velocity and to the characteristics of the granular mateials. Among these equations dP 2—c _-BV 8 dn I] where B is the function of porosity and C is a constant. Gaffney and Baird (1977) evaluated the resistance of bell peppers to air flow, and formulated the following equation: 32 _ . 13v L" [9] toexpressthe straightlinesinthelog-logpaperwithBasaconstantfardiscreteairflow ranges. Both Equation [8] and [9] are also special cases of Muskat’s equation. BenandChafity(l975)modifiedErgun’sequafiontorelatepressuredmptoair flow velocity and grain bulk density. Their equation % . A+BV+CV3 [101 is a second order polynomial function, whee constants A, B and C are functions of air flow velocity range (ml min) and porosity. Inanalyzingtheairflowresistanceofshelledcominhofizontalandvetical directions, Kay et al. (1989) used an equation like Equation [10] to represeit the resistance data. They concluded that air flow resistance in the horizontal direction is 58 and 45 percent of that in the vertical direction with air velocity ranging from 6.0 to 28.6 and from 6.0 to 0.8 m3/min/m’, respectively. They attributed these differences to the anisotropic characteristics of the shelled com. Considering that Ergun’s equation is limited to spherical particles and Ieva’s equation requires two coefficients, friction factor and shape factor, to be determined, Chandra et al. (1981) modified Leva’s equation to fit their data by using dimensionless analysis based- on Buckingham’s 1- theorem. Their equation has the following form: 33 tr, :- r;”(1851r,+1.7e§) [11] where r, isthepressuredropnumber, rrzistheReynoldsnumberand r,istlreporosity number. They reported that theprediction equation correlates theerperimertal data with a mean deviation of 10 percent. Haqueetal. (1980) meastuedpressureinabedofcornmixedwithnonuniform distribution offinesand formulatedabasicequationrelatedpressuredroptoairvelocity andfinematerials.TheirequationalsohasthesameformasthatofMuskat’s: = (A +c1’.)v-t13v2 [12] 84% whereA,BandCareconstants,andf.isthepercertagedistributionofthefine materials depending on radial and axial coordinates in a cylindrical grain bed. 2.4.3 Shedd’s equation During the 1940’s and 1950’s, Shedd conducted a series of experiments to determine the resistances to air flow of various grains. The results were preseited in the log-log scheme and these have been adopted as an ASAE standard since 1948 (ASAE Standards, 1988). Shedd (1945) firstattempted to use velocity V as a function ofpressure P to represent hisresultsforsarcom.Aftertestsusingothergrainsandseeds,hesuggested thatthe 34 pressure gradiert be used instead of pressure drop (Shedd, 1951 and 1953). In one- dimeisional space his equation takes the form of: v - a[£’;] ’ [131 dn- wheeAandBareconstantsrelatedtotbefluidpmperfiesandthecharacteisfics ofthe media. When B 31.0, thisequationissimilartoEquation [2]inone-dimersionalspace. Shedd (1953) also observed that the curves in the log-log scheme are convex upward. He indicated thattheaboveformula may fitthecurvesforonlyanarrowrange ofvelocity, beyond which the calculation according to Equation [13] may induce a considerable error. Staley and Watson (1961) conducted a test on the resistance of potatoes to air flow. This was the first attempt for determining the air flow resistance with large farm products. The experimeital data were plotted in a log-log scale and an equation like Shedd’s with A = 345 and B = 0.562 in English units was suggested to describe the curves. Staley’s data have beer incorporated into the ASAB standard (ASAB Standards, 1988). Wilhelm et al. (1978 and 1981) presented experimental data for snap beans, southern peas and lima beans in the same form as Shedd’s chart. They stated that a dimensionless pressure parameter, p = (AP/pJIXng), used in an equation will produce a better correlation between the velocity and the pressure drop, where AP is the pressure drop, H is the depth of the bean in container, gc is the gravitational constant and g is the gravity acceleration. Thus the following equation was put forward: 35 logV - O.49l2(logp)+2.1172 [14] Actually this equation is a form of Shedd’s equation in logarithmic form. Calderwood (1973), Akritidis and Siatras (1979) and Farmer et al. (1981) performed testsonmugh,bmwnandmilledrice,pumpkinseedsandbluestemgrassseeds, respectively. They also presented their data in the form of Shedd’s chart, which fit the expeimental data well. Grama etal. (l984)studiedtheresistancetoairflowofamixtureofshelledcom and fines. They observed the relationships between air flow velocity and pressure gradient under different levels of fines instead of different levels ofgrain bed depth. They found that air flow resistance of shelled corn increases when the fine material is addedandtheincreaseinairflowresistancebecomesgreaterasthesizeoffinesis decreased. Their results were given in a chart similar to Shedd’s. Jayas et al. (1987) enployed Shedd’s equation to match their data when rapeseed: and foreign mateials were used as media. They noted a difference in the resistances between the horizontal and vertical directions. The resistance for horizontal air flow direction was 60 percent of that for vertical air flow direction according to their report. 2.4.4 Hulrill’s equation Hukill and Ives (1955) recommended that pressure gradient be expressed as a logarithmic function of velocity. Their equation had the following form: 36 |% . AV” 1 dn ln(1+BV) [ 5] This equation fits Shedd’sdataverywellwithonlysmalldevian'onsforairvelocities from0.6l rrn/minto12.2m/min.Butdataforearcorndonotconformtothis expression. Equation [15] was later adopted as an air flow resistance equation in ASAE Data: ASAE D272.7 (ASAE Standards, 1988), with constants A and B given for particular grains. Todete-minethepressuredropinagrainbed, Spence(l969) tooquuation [15]as the mathematical model and suggested that two steps be talmn. First, the velocity can be estimated by solving the linear Laplace equation through complex analytic function: v - R- sinhamw) [l6] 2W cosh(21rx/W)-l where W is the width of the bin, and R. is the source strength related to mass flow rate. Secondly, the pressure gradient can be obtained by substituting V into Equation [15]. In this way, he reported that the calculated results give reasonable agreements with experimental data for a single duct arrangement in the tested system. Haque et al. (1978) modified Hukill’s equation based on their expeimental data for air flow resistance of corn containing various levels of fines: 37 _ av2 + _ 1 I% whereKvisafunctionofvelocityandLhasthesamerneaningasthatinEquation[12]. This equationalsohasbeenadoptedasanASAEstandard (ASAE Standards, 1988). 2.4.5 Sheldon’s equation Sheldon et al. (1960) investigated the resistance of shelled corn and wheat to air flow ranging from 0.003 to 0.3 m’lmin/m’. They found that the following equation correlating pressuredropPandairvelocityVcanbeusedtorepresentthestraightlinesinalog—log scale: P = Avn [18] where A = prlD and B = 2.0. The friction factor f, is also a function of velocity and porosity, while constants D and H are the equivalent diameter and height ofthe test bin, respectively. Osborne (1961) determined the resistance to air flow of grains and other seeds tint are commonly grown in Britain. He presented the data by using the static pressure as a dependent variable and the velocity as an independent variable. All the curves pass through the origin of the Cartesian coordinates, so the relationship between static pressure and velocity can be expressed as Equation [18]. 38 Later, Lawton (1965) measured the resistance to air flow of some agricultural and horticultural seeds and employed Equation [18] to establish the relationship between static pressure and velocity. Husain and tha (1969) and Nellist and Rees (1969) also used an equationsimilartoEquation [18]topredicttheresistancetoairflowofthreelndian varieties of paddy rices and soaked vegetable seeds, respectively. Rabe and Currence (1975) developed an equation to include the effects of velocity, moisturecontentandbulkdensityofdryalfalfaonstaticpressure: lnP - -3.5896+0.0005m°p.*0.0149pb+12351(an) [19] wherepbis thedrymatterbulkdensityandnuthemoisturecontentthhefirstthrce items at the right side of the equation are merged into one coefficient and the logarithmic function is changed into exponential function, then the above equation can be simplified as Equation [18]. Neale and Messer (1976 and 1978) investigated the resistances to air flow of root and bulb vegetables and leafy vegetables, respectively. These vegetables have relatively large size compared with that of grains and seeds, yet the pressure drop and the velocity still follow the same relationship as given by Equation [18]. For potato, they gave B = 1.80 when the units for pressure were mm W.G. and the units for air velocity were m/s. Constant A was calculated for different P and V according to Equation [18]. 39 2.4.6 Bear’s equation Bear (1972) reviewed various equations used to describe the nonlinw motion of fluid flow through porous media, except the equations that were used to predict the characteristics ofairflowthroughfarmproductbeds. Hedividedtheseequationinto threegroupsaccordingtothestatusofcoefficientsintheequations.Ingroup l, the coefficients were not related to any specific fluid and medium properties. Group 2 contained coefficientsmoreorlessrelated tofluidand mediumpropertiesandincluded unspecified numeical parameters. Group 3 was similar in nature to the group 2, but included the definite numerical parameters. For flow through isotropic porous media under steady state, most of the equations express the pressure gradient as a function of velocity and have the sanne form as that of Muskat’s. By combining Darcy’s law and the continuity equation, Bear (1972) derived a partial differential equation for incompressible fluids (p = constant and u = constant) and for inhomogeneous and anisotropic porous media: r1 +:—::1 12+: :1 1.1%] For inhomogeneous and isotropic porous media, the equation had the following form: a 51’ 3 3P 3 OP 8 [21] 3; [K3] -1-.W [KB—i] +.a_z. [KE] 0 1 For homogeneous and isotropic porous media, Equation [21] reduced to the Laplace equation: at? 32? its ax2+ay2+az= ' 0 [22] In this case, the pressure distribution is purely related to the geometry of the field. Bear (1972) stated that Equations [20], [21] and [22]-can be applied to both steady flow and nonsteady flow of an incompressible fluid for particular boundary conditions. For example, under the nonsteady flow condition the variation in time may be introduced through time-dependent boundary conditions. 2.4.7 Brooker’s equation Based on the analysis of Shedd’s data and on the consideration of the nonlinar characteristics of air flow patterns, Brooker (1961 and 1969) modified Shedd’s curve by using several straight-line segments, each of which has its own values for the constants oannd'BinShedd’s equation. 'I'hiswasaveryimportantsteptakentowards the approximation of the curve and the application of Shedd’s Equation (Equation [13]). In a two dimensional space, Brooker (1961) decomposed the pressure gradient in the normal direction into that in the X and Y directions: [22]” . [2]”.[93 ’ 1231 an ax 3y He also related the velocity components in the X and Y directions to that in the normal 41 &. aP/ax V. aPlan V _aP__/__ay V V aP/an [24] The following partial diffeential equation (PDE) based on Shedd’s equation and the theory of continuity for flow under steady state was developed: [121 2121 223111— 11212222 211-21]: :2 where m - (B-l)/2. 2.4.8 Segerlind’s equation Segerlind (1982) suggested that Brooker’s equations are not applicable to the analysis of the nonlinear air flow problem, since the variations of coefficients A and B wee not taken into consideration in Equation [25]. Considering botln coefficients A and B to be functions of the coordinates, he proposed the concept of granular permeability, K0. Using Equations [13], [23] and [24] and the continuity equation, Segerlind (1982) developed, in the two-dimensional domain, the following equations: 42 a tip +3 ap . alga] Eli‘s] ° ”6’ whereKoisalsoafunctionofthepressuregradient: B-l +-+[121’12’1’1" ‘2“ Equation [26] has the same form as Equation [20], but the coefficient K0 in Equation [26] has a different physical meaning. It is closely related to the local pressure gradient and the coordinates. The numerical value of K, can only be determined by experimental methods. Segerlind (1983) systematically analyzed the various forms of presenting experimental data on resistance to air flow. He suggested that Equation[l3] be adopted for describing experimental data and all velocity-pressure gradient data be presented similar to the technique used by Brooker (1969). It means that to express Shedd’s equation (Equation [13]) adequately, coefficients A and B must be assigned different values for various ranges of pressure gradient and air flow velocity. These suggestions togetlner with Equations [26] and [27] not only point out a way of generalizing the experimental data, but also imply a method for describing air flow through granular particles. 43 2.4.9 Other equations Grainstomgeshavebeenusedcommeciauysincednel930’s.1hefimtpapeonthe behavior ofairflow throughgrainbedswaspublished by Stimiman etal. (1931). They conecteddamonmeredmmairflowthmughmughficeindeepbinsmdpresented the data on a log-log scale described by the following equation for static pressure in the range of 250 to 1,000 Pa: v .. KHC [23] whereKandCintheequationareconstantsunderspecifiedpressure, andHisthedepth of the grain bed. Hall (1955) developed a relationship between velocity and pressure for bed depths less than 0.3 on. He used Shedd’s data (Shedd, 1951 and 1953) to do the analysis but employed an equation similar to Stirnniman’s to fit the data. Kelly (1939) obtained air flow resistance data on wheat, expressed by velocity and pressure drop for the different wheat depths: v = KPC [29] where K = 4005 and C = 0.5 in English units. Henderson (1943 and 1944) investigated the resistances of shelled com, soybean and cats to air flow, and recommended a general equation, similar to Equation [29], be used to explain the relationship between air flow velocity and pressure drop in the tested beds. He defined K as a function of grain bed 44 depth.Whenconductingtestsonsoybeansandoats,henotedthecuwatureofthefines inthelog-logscale. BunnandHuldu(l963)wnductedanexpeimentforairflowfluoughstedshoLBy mnecdngmedamfmvafiouspomsifiesofthestedshmbedfiheydevdopedequafions for both linear and non-linear flow. Under the linear flow condition, the equation was expressed in the exponential form: 6P _-Aexp 8n l: Obviously thisequationcannotbelinearizedwithrespecttotheparametersAandB. So av: _1 [301 ap/an it is impossible for Equation [30] to be fit to experimental data by the standard least square method. For the non-linear flow condition, they also used the theory of continuity and decomposed pressure gradient and velocity in the same way as done by Brooker (1961) to formulate a partial differential equation. They believed that if the parameter A is first established by using Equation [30] then it is sufficient to use their PDE to predict air flow patterns in a nonlinear flow system. 45 2.5 Representingfluidflowthroughporousmedia Fluid flow through porous media can be represented by iso-pressure lines, velocity distributions, streamlines, traverse time, volume flow rate, etc. These patterns are usually obtained byspecifyingindnecoordimMofaconfinedspacethecorrespondingdata, such as pressure and velocity values, which may be obtained from experimental or calculated results. . Collins (1953) used expeimental data from a grain dryer with an inverted U-shaped duct to develop the iso-the'ms, contour lines of equal moisture content, iso-pressure lines and streamlinesas showninFigureGa, b, candd, respectively. Thiswasthefirst such attempt using pressure contour lines, streamlines, etc. to delineate the air flow patterns witlnin a grain bed. Hukill and Shedd (1955) introduced the concept of traverse time into the representation of air flow patterns. According to their definition, the traverse time is the length of time the air takes to pass through the grain. They suggested that the traverse time be used instead of cfm/bushel to express the effectiveness of ventilation. By using this concept, they were able to draw equal ventilation lines for nonlinear flow in an oat drying bin (Figure 7) and corn drying bin. Hall (1955) proposed a graphical method for determining air flow at different locations in a grain bin. He also suggested that air flow rate per accumulated bushel of grain (cfm/acc bu) be used as an alternative measurement for air flow in a grain drying bin with a non-rectangular cross-section. 47 23?. . PRESSURE READINGS “" FLOW LINES ‘_" LNES OE EQUAL PRES”! LINES OF EQUAL VENTILATION Figure 7. Air flow patterns in the section of an oat drying bin. Source: Hukill and Shedd, 1955. Brooker (1958) analyzed lateral duct air flow patterns in grain drying birns. He revealed that air leaving a lateral duct will travel in various paths which are dependent on the structure of the bin and the duct. He used experimental data to plot the air distribution patterns for a grain bin with an inverted catenary duct, 11 semicircular duct, an on-floor rectangular duct having side openings and an inveted V-shaped duct having bottom openings as shown in Figure 83, b, c and d, respectively. The duct spacings were 1.22 m on center for Figure 8a, b and c, and 0.61 m on center for Figure 8d. 4m ,' eon-snu- 2 \\\\\\§. %\v:ov:v ‘ l ‘ é “MM/WM; 052:2' 2 2€I\ A: ~ 033! ’////////// MW0.0.0M \\‘kwfi WAYAYA ”Ill/Ill am: 292920 2' ”A? A A Ill/III” l!’ M“ Van.“ NW” MIA AQW/ ///////K§8Y" 623‘.“ “6,33,; WWW/”Mm JAM .-\:I \\ ‘ are“: ZWfltxg'I 23mg a WX’WM‘QIOX Ogmfi memmw ’7/ g n mrzmm Mir..?v‘v\€ g \\\2 $2933 2e91/JWW/ -AAA‘K’A 1 W43... [WW/Z. - '¢e¢6;6;6;0“\\\\ \: 0.252.??th 32.22.22. "ax \\ . 9; as m :33. esfik \\ manta iW/ YAYYAYA AA. .A reams“ \\\\\‘\\\\\\'O'0191026W/ll/IAYAYAYAYAYAYYAYA\\\W W \VIOZ OXOZYIY/I/l/llll ////""'A'A9 \m \\\m\\\sz x0; ////// W/nAAA’s. YAA\\\\M M\\ \zwzz' W16. A63.\\W I MW» ‘g‘sz ’WI} .A.x.A.A.“\V®\ \\ . \\\‘\ g WWAWW >292 9329203, .>\\\\\\® g lllllllllhs M\\\\M mmW.—-'a—-M *-—— gm... .22...." " ....—._{L.-='-_?W ....... may Tm” View \ .3‘A’A A Aer/MW W 420520 ”A '.»A A.A\\\\\ \\ Kb W/flflkeeee" €\.§\\§&\\\ bfifl/AWWAA A\\ \\W WWW/fl" 9293:3323. s\\\\\\\ \\\\\\\\‘\\ 'A ’/////////////////Z //.O202OA§AYAA.’ A 3\ ‘W\ w///////t////////3'§?;?A9: ' ' ”XWV§ HAW WWW/133331193 Ages '¢\\\\\\\W\\\\\ \\\\\\\\ AW/M/th §W£§A Yof‘2\\X\\&\\\\\\ \\\\ \\‘§\ 3.. \\\”? 92022 "”{i/ ”WV/124A “6?: A?A?A? \\\.“x 2%,? 0'0'051’4717/l/ A'A'A'A :Aflg' A? \\\\\\\\\\\\\k\\\\\\'\\-‘=:=2.- 2 - - 13“,: Infill]. (II/11127:: " “I- 7/4’ Figure 8. Air distribution patterns for different duct shapes and spacings. 0.61 on Source: Brooker, 1958. c. On-floor rectangular duct with duct spacing = 1.22 m a. inverted catenary duct with duct spacing = 1.22 m d. Inverted V—shaped duct with duct spacing b. Semicircular duct with duct spacing = 1.22 m 49 Brooker (1961) applied the finite difference method to the calculation of air flow parameters for a grain bin with a rectangular duct having side openings. He plotted pressure contour lines for four different values of B in Equation [13] and compared them with those obtained from experimental data. The pressure patterns determined by the numerical method with B = 1.0 and that from experimental data for air flow through a wheat bed are shown in Figure 9a and b, respectively. He showed that for a rectangular duct with side openings the largest spacing between iso-pressure lines is located at the lower corner of the bin cross-section, where the velocity is relatively lower. Ives et al. (1959) experimentally studied two dimensional, nonlinear air flow patterns and drying patterns for grain. They found that all air flow streamlines are straight and parallel above grain depth H which is 0.5 times the distance L,l between the ducts. They also reported that there exists a stagnation point midway between ducts where there is practieally no air movement and all ventilation fronts may be considered forming asymptotic tails anchored to that point. Based on Hall’s method (Hall, 1955) of analyzing non-parallel airflow, Boyce and Davies (1965) investigated the effect of lateral duets with four different exhaust areas on the air distribution within a barley bed. They indicated there was a large variation in the air distribution in the grain around the duct depending on the air exhaust area. Figure 10a, b, c and d show iso-flow lines (left) and iso-pressure lines and streamlines (right) for percentage duct opening area to total duct surface area of 100%, 68%, 39% and 13%, respectively. The volumetric flow rate was 0.34 m’lmin for all the ducts in their analysis. Note that the unit for iso-flow was ft’lmin/ft’. I “8 u 0” ° ”' o L— 0 Z 8.. I... 32. 3,. —- 2; .- 9 4 on an «'0 .3. r- 9-5- 9.0 5.0 I“ a. ' ' Q C l‘” -...__ __ O 7 ’°' 7:0 130 .130 0! ._._ _.._._ - , 0 ’ fl 9; 01.0 ”5 IO an I”: a”: .0... . . '1 I” I)” I.” It?) ....___., __ __ l3 _ _ I. L ¢ 00" I030 ‘03. I 5 a I, ’M ’N .'. / '. .q. ’ \_‘ /;"‘\ "’ “4K “ / A\ “ ‘ .’\ t .' ' 2 ' #fi \’ m. '0; .010. 'Q/ : : \;‘\\ o’ 1 .| . \ ‘-~ I; #- Aha [w E 2000' g .\ \\ H a). '02. L!!! i. 4' \ \ 50 on 0:" on.“ on.“ 00.09 0.2 03 on. . 03's: 0!): also 0 l no.0 ago: ago: 03” __ 0.5 00 out «'39 0d» ooh 0.7 0.?” or." 0‘?! 07.10 0.0 0.! out ”'21 ”in ciao LG I“. I”, I»? 'm ' ' L2 no. 12.“ L19 It.” 1.! [4 to. u 1.00 «it LS I... I}... . l I!” l.‘ I0" . 00.00 . .000 I0” -—_—___/—‘\ M 18 I?” ’ I : z a ’ I?“ / 1' ' o ' he. ./.'.;’ .d 2.000 .0 gm nu um é» ‘I Figure 9. Pressure patterns established by numerical method and by experimental data. Source: Brooker, 1961. a. By numerical method with B = 1.0 and P‘l = 500 Pa b. By experimental data with I"‘ = 500 Pa 51 _ § 2 u 88 85.8 8.6 .88 9 8.3 weenie 8.6 emacoobm .v .5 an n 8.3 03% 8.6 :39 9 8.3 wficoqo 8.6 ems—580m .o a» me u no.8 3855 8.6 .89 9 «an mew—one ~26 emacoobm .2 me can no.3 8855 82. .89 9 98¢ mcfioae 8.6 amass—om .a $3 6325 98 83mm "venom .mmcmcoao 8.6 595:6 .8 85—883 28 8:: Banach—6% doe: Becca .3 95mm .85 u 6.3 . 5.3 .26 on .J-__o 70.0 .OV 52 Barrowman and Boyce (1966) conducted experiments in a barley bed to determine the effectsofduct spacingandopening, grainbeddepth, andairflowrateonairdistribution and pressure losses. These experiments are the continuation of work reported by Boyce and Davies (1965). For comparison, they employed the R ratio concept suggested by Rabe (l958)toexpress theduct opening area, inwhichRisdefinedastheratioofopen area of duct system to floor area served by duct system. For a perforated floor, R = 1.0. They found many duct systems have R values as low as 0.11 to 0.09. They recommended an R value of 0.25 and concluded: 1. Increasing theduct spacing willreducetheRratio. In thiscase, ductpressurehas tobeincreased to maintain theairvelocity constantintheparallel flowregion. This will in turn increase the duct entrance air velocity. Therefore, air distribution in the bed will be more uneven. 2. Reducing the duct opening area only has a slight improvement in air distribution. 3. Increasing the grain depth will improve air distribution in nonparallel flow region. 4. Air flow rate will affect the position of the iso-traverse time line, but will have only a slight affect on their shape. Hohnor and Brooker (1965) used an analog method to predict the shapes and positions of the cooling front in a cylindrical grain bin with a cross-flow ventilation system. They noticed that the accuracy of this method depends on the deviations of the prototype system from a Laplace field. For the flow rate commme used in grain ventilation systems, they believed the analog method to be quite accurate. Brooker (1969) again used the finite difference method to predict air velocity distribution in a grain bin with a rectangular duct having side openings as shown in 53 Figure 11. He noted that the air velocity is uniform in the upper portion of the grain bin. I '13. / V >8L5 Y'all-5 I % ILI‘I D 0.3 ”’1' ’ 0'52. V>2|$ % s ms I .02. lawman § \\ I 0102 2L8M>I°£ .1 IO! ll 3 >V NOD II .10! x. o > V > o. n It)“ 0 l > v ’ n w I“ 00.0>v>d s MIC? Figure 11. Velocity distribution in the lower portion of the bin for three duct pressures. Source: Brooker, 1969. a. DuctpressureequalsthSOPa,andunitofVisfprn b.Ductpressureequalst0500Pa,andunitofVisfpm c Duct pressure equals to 750 Pa, and unit of V is fpm scat: o o s' a a 35 ‘5 SO a s: $0 63 Figure 12. Calculated pressure patterns expressed as percentage of duct pressure. Source: Brooker, 1969. a. Duct pressures: 250, 500 and 750 Pa, B = 0.628 to 0.768 b. Duct pressure: 500 Pa, B =1.0 55 Brooker (1969) also claimed that when the iso-ptessure lines from the numerically calculated values are plotted as percentages of the duct pressure, the geometries of the iso-pressure lines remain the same if B in Equation [13] is unchanged. Figure 12a is the calculated pressure patterns expressed as the percentage of duct pressure for three duct pressures of 1.0, 2.0 and 3.0 inch water (250, 500 and 750 Pa) with B = 0.628 to 0.768. Figure 12b is the calculated pressure pattern for a duct pressure of 2.0 inch water (500 Pa) with B = 1.0. From Shedd’s data, the B value will approach 1.0 when air velocity is low. In this case, he recommended the use of Laplace equation as the governing equation in the calculation. 3°1- so '1. 50‘! as _2L 1L____. ..____IQ_—_——t _____.___I.D._.‘ II——_. as A . 5 to A: IL rs at I U 9 I I I 2250 Pa '2/ 5500 Pa ’ :750 Pa a b c Figure 13. Pressure patterns obtained from experimental data for different duct pressures. Source: Brooker, 1969. 56 However, Brooker (1969) also found that the pressure patterns obtained from experimental data are different for duct pressures of 250, 500 and 700 Pa as shown in Figure 13a, b and c, respectively. This is contrary to the result obtained from the numerical calculation as stated above. Iindal and Thompson (1972) used the same equation and method as that used by Brooker (1961 and 1969) to analyze two-dimensional air flow patterns for long triangular shaped piles of grain sorghum with a rectangular lateral duct through the center of the pile. The top and two sides of the duct were perforated. They developed a numerical procedure to find the flow streamlines, which together with iso—pressure lines are shown in Figure 14a, b and c for grain repose angles of 25°, 35° and 45°, respectively. They concluded that the duct size greatly affected the total air flow rate for a given pile configuration, but the repose angle seemed not to affect the total air flow rate. They also pointed out that increasing the base width of the grain pile decreased the total air flow rate. Pierce and Thompson (1975) extended the work done by Jindal and Thompson (1972). They intended to predict air flow patterns for a conical shaped grain pile with a rectangular duct through the center. The effects of the duct size, repose angle and pile diameter on air flow rate were the same as that mentioned by Jindal and Thompson (1972). The air flow patterns from their ealculatcd data is shown in Figure 15. The shaded area represents the region near ground, where the spoilage of grain is most likely to occur. The air flow rate for the lower shaded area, which accounts for 37 percent of the pile volume, is less than 50 percent of the average air flow rate for the whole pile. 57 MM ”ROY-Rnog an M' "M'lom 00‘. m ”WIVNZO‘ou I. “1 ”(Mum-IO coa- ooo- 'h ‘- IIMWUNIGIWIIYfl201IA All MT mm 01.00“ 000' —Mwa Liv-s Figure 14. Air pressure and flow path patterns for different grain repose angles. Source: Jindal and Thompson, 1972. COIN 33. ANGLE 0' .6708! DUCT DIAIIYIBS 10% OF I’M! POLE OIAIITER ISO'PIISSUII LINES ----'- AII'LOW PATH8 PIIFOIATEO RETAINING WALL Figure 15. Air pressure and flow path patterns for a section of a conical shaped pile. Source: Pierce and Thompson, 1975. 58 Based on Broolner’s nonlinear partial differential equation (Equation [25]), Marchant (1976a) first applied the successive over relaxation numerical method to the solution of this equation to estimate air flow patterns within rectangular and cylindrical hay bales. Late, realizing the disadvantages of the finite difference method in solving the partial differential equation, Marchant (1976b) employed the finite element method to solve the following equation in two dimensional space: a _16P)+_6_16P):0 [31] 3x (as; ay‘r‘ii in which K is a function of velocity. He found that the ealculated values were very close to the experimental data for linear air flow. He concluded that the finite element method can cope with any geometrical shape and any correlation of pressure gradient with _ velocity. He plotted iso-pressure lines for a rectangular grain bin with lateral ducts having different shapes and openings. He compared these figures with Broolner’s experimental data (Brooker, 1958) as shown in Figure 16a, and with Barrowman and Boyce’s experimental data (Barrowman and Boyce, 1966) as shown in Figure 16b and c (the dotted lines represented the elements). He noted that the calculated pressure contour lines overlapped those obtained from experimental results. Using Equation [12], [23] and [24] and the continuity equation, Haque et al. (1980) developed a partial differential equation and used the finite element method to calculate pressure and velocity values and to determine the direction of air flow movement for any point in a conieal-top cylindrieal grain bed with a perforated floor. They claimed the pressures ealculated numerieally agreed well with the observed data. 59 ' 1 cc 0 a J 3’9 I . I 0,, : .I H ! s-s f 0.; : l 1 34 | I fl “ I fl : 1 0-3 ! ‘4 J 32 J o-s l l JI M J. M I as 1 I 1 ,l E ]- " J " 4 ca . i . ; i ----------------- J ---------------- j 9.? I I ’Q I 2‘ jI E I H | 14 J 0" 1 fl 1 1 I E 32 J E 3‘1 ' 9’ l I ' e In E . fl s I 9 fl "’ ~ 3-0 J " 2.0 I E 'r """""""""" l \ ‘\ t'. ’ \“ 1 as a I \“ '..‘7| :4 gv I r \V l “.‘ 0" ‘\ " M ‘ 20 3" ‘:A:‘ ' “ t a ’." t.- __ .... 1 _ b.-;- 'v .‘0. E ' E W. :‘s.: I .. ' -- a -3 :9 . [o -.-$ --+-. I ' '3' : n I 0.451“ ff 0°46!!! a b c Figure 16. Calculated iso-pressure lines in a grain drying bin with different duct shapes and openings. Source: Marchant, 1976b. a. Calculated iso-pressure lines overlapped those obtained from Brooker’s experimental data. Duct opening = 0.2 m. b. Calculated iso—pressure lines overlapped those obtained from Barrowman and Boyce’s experimental data. Duct Opening = 0.051 m. c. Calculated iso-pressure lines overlapped those obtained from Barrowman and Boyce’s experimental data. Duct opening = 0.152 m. 60 Lai (1980) modified Ergun’s equation (Equation [4]) and used the method of lines to convert a nonlinear partial differential equation into a system of ordinary difl‘erential equations to solve three-dimensional air flow problems in a cylindrical bed with two different porosity distributions. He found that the pressure distribution showed significant variations at the air entrance because of the nonuniform entrance velocity. He also observed that a bed with nonuniform porosity will generally display a lower pressure drop than will a bed with uniform porosity, even ifthe average void fractions are equal in both beds. Segerlind (1982) used the finite element method to solve Equations [26] and [27] for a rectangular bin filled with shelled corn. His computed results both for linear flow and nonlinear flow were very close to those given by Brooker’s calculated results (Brooker, 1961) and experimental results (Brooker, 1969), respectively. The pressure contour lines: in the entrance region with a duct pressure of 3.0 inch water (750 Pa) are shown in Figure 17. The values of the upper and low comer nodes were incorporated into the impermeable boundary conditions in his calculation. Khompos (1983) and Khompos et a1. (1984) applied Segerlind’s equations to air flow problems in a three-dimensional domain and also employed the finite element method to analyze air flow patterns for cylindrieal grain bins with different duct shapes and locations. The typical pressure and velocity distributions for a Y—shaped' duct with a grain depth of 9 m are shown in Figure 18 and 19, respectively. 61 Figure 17. Pressure contour lines in the entrance region. Source: Segerlind, 1982. Smith (1982) used the finite element method together with a frontal solution technique to solve Equation [31] for hay and grain beds in three-dimensional space. He thought that K can be taken as a constant for linear flow with low air velocity. Fornonlinear flow, he held that Equation [13] can still be used if K is expressed as a function of velocity. He believed that his method can be used to obtain the solution for air flow problems with reasonable accuracy. But he also found that the calculated air velocity is less accurate than the pressure value and most of the errors arise in regions with high air velocities. 62 .9 .33 .8955— 6058 .82. 83.....-» .8 Eugene 838$ .2 8:3... ngxctzx 7535+...“ .. ”an.“ .+ a .r a w L. 1.. E can T". .8 To r/ l\ 0 Sm an; 4. .3. J.“ m Jam-r 8n n. Y... 8n .n. no~ ..1 1.05 .l '0! 0°! .6 .0 0L .9 '9 0' .8 oz 0 ('HISIXU"A 'l! .mwefi £8899 "venom .eoao Bazaar 8.. Sesame b_oo_o> .2 use... ..:.o~x¢t>ax ~.:.m~x¢tx o a u a o o a u _ o lb in [ID pa .0 63 '3 '9 (“HISIXB‘Z I U I 'L '9 '3 U ’0 V '6 '01 ' 0L 09 cs 0’ .6 oz 0‘ ('HISIXU-A '0! '1! 64 Miketinac and Sokhansanj (1985) employed an equation similar to Equation [31] as the governing equation and used the finite element method to ealculate the pressure distribution for the grain bed as outlined by Brooker (1969). To deal with the difficulty of obtaining accurate numerical solutions for the entrance comers of the rectangular duct, they proposed that a refined mesh be used near theair duct. The resulting pressure patterns for air flow in a Laplace field are very close to those calculated by Brooker (1969). The calculated pressure patterns for nonlinear flow are shown in Figure 20a, b and c for duct pressures of 250, 500 and 750 Pa, respectively. The values of constants A and B in Equation [13] were taken from Segerlind (1982). Comparing Figure 20 with Figure 13 they found that differences existed, which they attributed to the inaccuracy of the experimental data. Later Miketinac et al. (1986) used the same technique as Miketinac and Sokhansanj (1985) to analyze the air flow patterns for several bins and floor configurations. The typical velocity vector for bins with semicircular and rectangular ducts are shown in Figure 21a and b, respectively. Chapman at al. (1989) applied the finite element method to the solution of Segerlind’s equations (Segerlind, 1982) for the analysis of air flow patterns in a grain storage with different duct distributions. They thought that the pressure pattern alone is not the best way for describing nonlinear air flow behavior, but that the isotraverse time lines as introduced by Hukill and Shedd (1955) can reflect the actual shape of the temperature front when it moves through the grain bed. The pressure contour lines, the streamlines and the iso-traverse time lines for flat grain storage with three circular ducts are shown in Figure 22a, b and c, respectively. Y-AXIS 65 Y-AXIS 8 Y-AXIS 8 \ w 82.5 I 100% .5 ,5 : 750'0 l X-AXIS a b C Figure 20. Pressure contour lines for grain bed with different duct pressures. Source: Miketinac and Sokhansanj, 1985. a. Duct pressure equals to 250 Pa b. DuctpressureequalstoSOOPa c. Duct pressure equals to 750 Pa ALLJ L14 LJJ LJ_LLJ LLJ-LL w—w "VI-w" IIIIIIIIII II III II III I I I I I \\\IIIII’IIIII ss\\\\\\‘ 0 O 0 t’IIIoo ’I’I’Ioso I’laaoeee ’40.... ii in. as\\\\\ 0§\§\\\ a 0 QOO§KKr ‘ \ L4 ILJ L14 LJJ‘LJ LivJ L14 I I I III I IIIII I I I I II II I I I I I I I I I I I I I I II II I I I I I ll’loo \\\\'\ \\\KK \ d’lloo ”Adena. ¥ Figure 21. Velocity vector fields for bins with semicircular and rectangular ducts. Source: Miketinac et al. 1986. 0'3 Is) I 6'3 1 67 3 duets Pressure camera O 5 IO ISII 20 23 30 0 2 4 III 6 8 IO '1‘ II II 1° I8 I8 3 CUCI‘ d: '6 Streamlines 1 ‘5 is : 14 J4 r2 ' - 12 ‘0 0 : .l ‘1 IO " 3 a . .0 a H . 55:: : :' .. 1 g H : 9 : :0. : 0. .' “ a :i: :: .- .° .- , g '12 e :. z: . a .0 .0 _‘ I. . . O .. O . . .0 ‘ n . Q . .. . . O . -‘ ‘ ' . . . 3: : .s -l " I 9 ° -- 2 5 ....0 4 3 “seat ' ' ' ' o .9 O 5 IO 15" 20 25 30 C 2 - rn 6 3 IS m It It Is I! I8 "' Bauer: J 5 I0 l- Isotreverss tomes - ID I I4 *- .t '4 l. I. .. -4 12 ‘l 12 I— 5 ct IO '- d to 43 r- -l 8 ’- 4 B . a N . ‘2 b ‘: as e3 D ‘ 4| 2 - 1 \ z o .i' I I v I t 1 I I 1 1 I g I I I I °.' I I 1 '1 I 0 JO 0 5 IO IS" 20 23 30 O 2 4 m 6 8 IO Figure 22. Air flow patterns for flat grain storage with three circular ducts. Source: Chapman et al. 1989. CHAPTER 3 ANALYSIS METHODS 3.1 Establishment of mathematical models 3.1.1 Modek of pressure and velocity distributions The equations governing air flow through the potato storage are derived from the basic physieal principle of conservation of mass and from equations developed through the analysis of experimental data. The principle of conservation of mass defines that, when fluid flow passes through a confined space the net out flow of mass from a volume should be equal to the inflow minus the decrease of mass within the volume. Following Eulerian description, Incropera and DeWitt (1985) and Sabersky et al. (1989) used the concept of a differential control volume to derive the continuity equation in three-dimensional Cartesian coordinates: 93 a ._ .1 .3. . 32 at 0x (pV‘) 8y (pv’) azo’v') 0 I 1 Suppose the fluid under study (air) is incompressible, _p = constant, then 68 avuavuav, . 0 _ _ _ [33] 6x 6y dz Equation [33] is the reduced form of Equation [32], or simply, divV = 0. The relationship between the air velocity and the pressure gradient for air flow through the potato storage can be expressed by Equation [13] (Staley and watson, 1961). In a three-dimensional domain, the pressure gradients in x, y and 2 directions can be relatedtothatinthenormaldirectioninthesamewayasinEquation[23]: 22’.2£’.2£’.22’ [34] an ax 3y dz Thevelocitycomponentsinx,yandzdirectionsalsocanberelatedtothatinthenorrnal directioninthesamewayasindicatedinEquationm]: < x g arr/ax BP/an <| | < , g ZIP/By V aPlan < g g BP/az [35] aPlan <1| 70 From Equations [13], [33], [34] and [35], a partial differential equation can be developed: axon] -a-,[Ko%— if] 514 Keg—Ho = [361 where the granular permeability 4112—1 «:11 1:— :11“ Equations [36] and [37] are actually the representations of Segerlind’s equations (Equations [26] and [27]) in three-dimensional space. ' Equations [36] and [37] can be used for inhomogeneous and isotropic porous media, because the coefficient K0 is a function of the coordinates, but not a function of the orientations of the media. These two equations have been successfully used to analyze air flow patterns in a grain storage (Segerlind, 1982, Khompos, 1983 and Chapman et al. 1989). They will be used as the basic equations in the present study for the calculation of the pressure distribution within the potato storage. After obtaining the pressure distribution, the velocity in the normal direction and the velocity components in x, y and 2 directions can be determined from Equations [13] and [35], respectively. 71 3.1.2 Models of streamlines and flow rates Todescribe fluidmotioninaspace, anirnaginarystreamlinecurveisintroduced. It is defined as a curve everywhere parallel to the local velocity vector (Greenlrorn, 1983 and Sabersky et al. 1989). By this definition, the relationship of the increment of the curve, dx and dy, and the velocity components, V, and V,, can be expressed by the following equation: dx V _ . _" 38 dy V, I I or -Vydx +V‘dy = O [39] In two-dimensional space the continuity equation (Equation [33]) has the form of flq-BL a O . [40] 6x By By introducing the stream function ‘1' and defining that lir 3‘1! ’ 0x I 1 Equation [39] is equivalent to 6‘1! 89 6x + By y I l and the continuity equation (Equation[401) remains satisfied. Because‘I'isafunctionofxandy,d‘I'canbeexpandedas d? at fldx-Ifldy [43] 0x by When ‘1' is constant, i.e. d? = 0, Equation [43] is the same as Equation [42]. Therefore, lines of constant ‘1' also represent streamlines (Sabersky et al. 1989). From Equation [24] vy 'a'P/ay' and from Equation [41] 3.x . - "’3? [45] V - B‘I'ldx R 73 it follows that gg§§+gg - o [46] Equation [46] means that the velocity potential (pressure, gravity or their sum) and the stream function are mutually orthogonal because their inner product is equal to zero. That is to say the streamline is always perpendicular to the iso-pressure line. Therefore, if the pressure distribution is known, then the streamline can be obtained from the orthogonal relationship of Equation [46]. As mentioned above, the streamline takes the direction of the local velocity vector, there is no flow across the streamline. The boundaries of the stationary solid surfaces and the symmetric surfaces are always the streamlines. According to these concepts the rate of fluid flow in the area between two streamlines will be constant. 3.2 Application of finite element method The finite element method has been used for solving problems of solid mechanics for a long time. During the past decade, the finite element method has found increased use and wider acceptance for the solutions of the equations governing fluid mechanics and heat transfer. The main idea of the finite element method is to change the continuous problem into a discrete problem which is represented by a system of algebraic equations. When a governing equation for a field problem can not be solved analytically, the finite element method may be the best alternative to solve it numerically. Generally the 74 solutions of Equations [36] and [37] can not be found by classical analytical methods, exceptwhenKoinEquation [37]isequaltoone.lnthelattercase, Equation [36]is reduced to the Laplace equation (Equation [22]) which together with certain boundary conditions can be solved analytically (Churchill and Brown, 1987). In most cases, the only way to solve Equations [36] and [37]is to resort to numerical methods (Hariharan and Houlden, 1986) - at present, the finite element method. The procedures of applying the finite element method are outlined in the following sections. 3.2.1 Applying the Galerkin method The Galerkin method is a member of the larger class of weighted residual methods. It makes the residual of an equation for a certain solution orthogonal to the interpolation function of each element. Thus the inner product of the residual and the interpolation function equals zero (Fletcher, 1984 and Ortega, 1987). In the Galerkin method, the elements are isoparametric because the interpolation function (also known as shape function or weight function) used to describe the coordinate transformation is chosen from the same family of the trial function which is used to represent the dependent variables. In this case, the convergence to the exact solution can be secured as the node number tends to infinity (Fletcher, 1984). The objective of applying the Galerkin method to Equation [36] is to reduce the partial differential equation to a system of algebraic equations. According to the inner product law, the weighted residual integral equation has the following form (Segerlind, 1984): 75 was: —::1:1<:— 51:1s2— :11w whae[N]istbemwvecmroftheshapefilncfiminCuwdanmordimm.Fmflreabwe nonlinear formulation, it is reasonable to choose a twenty—node three-dimensional solid element. The row vector ofthe shape function for this element is given by [N] = [N, N, N, ...... N30] [48] The finite element approximations are X = [MN Y = INlly} Z = {Mil} P‘°’=[N1{P"’} [49] 3.2.2 Applying the Green-Gauss theorem The general form of the Green—Gauss theorem (Sneddon, 1976 and Pearson, 1983) is 3Q 3R 3T 3 + + 5 IV_[ art '67“ az _]dxdydz I (Qcosa RcosB Tcosy)dS I 0] where cosa, c033 and cosy are the direction cosines of the curved surface 8. By using the chain rule and applying the Green-Gauss theorem to Equation [47], thrs second order integral can be changed into a first order integral and surface integrals. 76 Note that by using {R“’} = {0} and the homogeneous boundary conditions (Churchill and Brown, 1987), Equation [47] will be reduced to mantra} -_{0} [511 where [16"] is an element stiffness matrix, [Kw] - [ VlBl'lDIIBIdV [521 The row vector [B] contains the first order derivatives of [N] with respect to the coordinates of x, y and 2. It has the form of as, an, an, 31s,, 3' '3? Tx' "as" an, aN, an, an” [Bl - ...... _ 6y 6y 3y 6y [53] aN, an, art, 8N” _W Ta” ‘52' Tad The matrix [D] is a diagonal one with the K0 values defined in the diagonal. The coefficient K0 is a constant value for a particular step in the calculation, but must be updated at each iteration step. 3.2.3 Jacobian transformation Before Equation [52] can be solved, the physical coordinates (x, y and 2) related to all the variables should be mapped into the natural coordinates (S, n and 1'). For the twenty-node solid element (Figure 23), the shape functions for these nodes in the natural coordinates are as follows (Kardestuncer and Norrie, 1987): N. = %(1 +5.10 +n.)<1+i.)(t.+r.+i.-2) i=l.3.5,7,13,15,17,l9 N. - %<1-is(1+ns<1+rg i=2,6,14,18 N. = 71-01604th i=4,8,l6.20 N, = §(1-e11 +5911 +11.) i=9,10,11,12 [54] where I. = 55$. n. = rm and 1'. = {if for node i. C 11 n ‘19 A18 V l /‘ 20 l 16 / 1 1 / L14 | 15 // I12 I / 1.11 I | / l I 14_____ =g l l 90 '7 1011 /L-————-—a-6- ————— 15 / 8// / 4 / / I '2 3 Firgure 23. Sketch of twenty-node solid element with natural coordinates 79 Byusingthechainrule,thepartialdifferentialformsoftheshapefunctionswith respecttonaturalcoordinatescanbeexpressedas aNi(£I’IID. aNi 6x +aNlay+ ONE 82 T dx d£ 6y d£ dz OE ”Hand? 31‘1th 3N dy aNtdz an d—x 611 fly 01) Oz 01; «Liana, 6N. as HaNtay 3“ 62 1551 T; as 3’; ay at 32 at Equation [55] can be represented in the following matrix form: "memo“ 'aNJ as W 6N3“ rm!) 3N5 _a’I 3 [I] W [56] 3N5“ rm?) 3N: . a: _ .797. where Jacobian matrix [I] is given by p In _ 2 Of 3£ By 62 3” E [571 _a_z 6: an MI? Q" [11' 2|? 343’ °’ ~23 'l‘husthematrix[B]inthenaturalooordinates_beeomes ' ammo“ 3f 3N 9 s [memo] = [1]" "5 " O a" [58] aNiG My!) . a: . where [1]" is the inverse of [I], -1 g adj [J] [59] [J] 'I—Tdetm The differential volume of the element, dV, is given by dV = dxdydz . ldetU] |d£dndf [60] After Jacobian transformation, the element stiffness matrix [16"] becomes the integrals in the natural coordinates (Segerlind, 1984), 81 [Km] = j _1‘ I _1' J _:l[B(E.n.D]'[D][B(£.mm Idetmldsdndr [611 3.2.4 Gauss-legends? quadrature Because of the presence of Jacobian matrix [J], the exact solution of Equation [61] is still very difficult. To evaluate integral Equation [61] numerically, Gauss-Legendre quadrature is introduced. The sampling points, 11, can be estimated from the formula: (Zn-1) = N, where N is the degree of the polynomials (Segerlind, 1984). For the present problem, the highest power of £, :1 and 1' within Equation [61] is equal to 6. Therefore, the choice of 3 sampling points for each of the variables is suitable. The locations of the sampling points and the weighting coefficients ean be obtained from the related reference books (Segerlind, 1984, Reddy, .1984, and Kardestuncer and Norrie, 1987). For three sampling points ft = 0.0 wi = 8/9 E; -'-"- 10.774597 w, = 5/9 The integral Equation [61] thus becomes a numerieal summation: 3 3 3 [KM] = 2: 2: 2 tf(s..n,.r.)w,.1 I621 M 5-1 h] where 82 f(£,.n,-.l‘g) ' [B(£.n.D]T[D][B(£.mO] ldetUJl [53] 3.2.5 Applying the direct stiflm method A single finite element is continuous in nature. But the continuity requirement for the discrete representation of the entire region can be satisfied only by assembling the individual element matrix [16"] into a global matrix [K]. This process is the direct stiffness method. For the present problem, the global matrix [K] is a symmetric, banded and positive definite matrix. At this stage, the partial differential equation (Equation [36]) is finally changed into a system of homogeneous algebraic equations which are represented by a matrix form: mm = {0} I641 The nodal pressure values ean be obtained by solving Equation [64] along with the prescribed boundary conditions. 3.2.6 Applying the Newton-Raphson method In determining the pressure and velocity distributions at any point, say A(xA, yA, 2A), of a cross-section of the potato storage, three sets of data are required: the element 83 number where point A is located and the coordinates of the nodes of this element, the nodal pressure values of this element, and the natural coordinates of point A. The Cartesiancoordinatesandthenammleoordinatesarerdatedthmugthuafion [49]ofthe finite element approximations which can be rewritten as PKEJM‘) ' [N]{X}-XA 3 0 F450,!) . [N]{Y}'YA O F3(£snsn s [N]{Z}'ZA '3 0 [65] Thus F,, F, and F, form a set of nonlinear algebraic equations. To obtain 5,, 11A and {A corresponding to xA, yA and 2A from Equation [65], both the successive substitution method and the Newton-Raphson method are available. Because the Newton-Raphson method converges quadratically, it takes only two iterations to reduce an error of 102 to an error of 10" (Finlayson, 1980, and Chapra and Canale, 1988), this method is chosen to solve Equation [65]. 7 By expanding Equation [65] in a Taylor series about the kth iteration and neglecting the second and higher order of the derivatives, the following equations are obtained: 31%“pr 1' g) Fifipvnkol’rk") 8 F,(£p’lgtl'r)* as (5...; -53) + 3F.(£..ng.l'g) , ’7 + aFiGp’lgs a) an \ to] #1..) a}, (as-m I661 where i = l, 2, 3. Because it“, n“, and I“, are expected to be the solutions of Equation [65], let Him. mm, (M) = 0. Equation [66] can be written in matrix form: 1min: Fr- [AL "tor 'flrt ' " F2 [57] _rpr'rg F3_ t where the Jacobian matrix [A], is given by F 3F, 3P, 3F,“ 0£ an 3! aF, Mi, ari, 5% 3n 35' 3F, 6F, 6F, _ 66 an a:_ k [A1]: = [68] The subscript It means that the related variables are evaluated in the kth iteration. Since . values of E, n and {are between -1 and +1, the selection ofzeros for E.» n. and 3'. is reasonable. 85 3.3 Compilation of computer programs The computer programs are designed mainly for establishing within the potato storage the pressure distributions, velocity distributions, streamlines or air flow path and flow rates. All the required information about the air flow patterns can be obtained from the analysis of these data. The computer programs were written in FORTRAN and run on the IBM Mainframe 3090 in the Computer laboratory, Michigan State University. 3.3.1 Calculation of nodal pressure values The program uses the procedures outlined in Section 3.2 to find the pressure distribution in three-dimensional space. The input data of the incidence matrices, including element number and nodal number, and nodal coordinates can be produced by automatic mesh generation programs (Ansys, Manual, 1987 and FIDAP Manual, 1989). The incidence matrices and nodal coordinates are different for different duct configurations, different duct spacings and different depths of the potato pile. The band width can be obtained from the information contained in the incidence matrices. During the process of ealculation, K0 was assigned a value of 1.0 at the first step, but was updated in the subsequent iteration steps. Detailed processes are shown in the flow chart in Figure 24. The output of this program was used as the main input data for the calculation of iso-pressure lines, velocity profiles, and streamlines and air flow rates in the selected cross-sections. BEGIN i INPUT ' of incidence matrices, nodal coordinates, band width, coeff. of A and B, and boundary conditions l INITIALIZATION of nodal value vector {P} and force vector {F} with zero i CALCULATION of matrix [B] and [B]T, Jacobian [J] and Idet [J]| normal pressure gradient «With, and granular permeability KG (KG=1.0 at first step) l FORMATION of element matrix [K‘°’] according to Equation [62] and Equation [63] l SUMMATION of global matrix [K] by using direct stiffness method l 87 MODIFICATION of global matrix [K] and force vector {F} according to the boundary conditions i DECOMPOSITION of modified matrix [K] into upper triangular matrix [U] by using Gauss elimination method i CALCULATION of nodal value vector {P} by using backward substitution method [ COMPARISON of AP: [Pm'PoIdI with a preset value 8. if AP>e, then substitute PM, into P0,, and run the iteration loop again; otherwise output PM, i OUTPUT of nodal pressure values l END Figure 24. The flow chart of the computer program for the calculation of nodal pressure values in 3-D space 88 3.3.2 Calculation of pressure and velocity for the selected cross-section . The main purpose of this program is to calculate the pressure distribution and velocity profile for selected cross-sections which are perpendicular to the center line of the lateral duct. The input data are 3-D nodal pressure values, the incidence matrices and nodal coordinates in 3-D space and in the 2-D cross-section, and the coefficients A and B. The transformation of the coordinate systems plays an important role in the investigation of the pressure and velocity variations in the selected cross-section. The output of this program will be used for plotting pressure contour lines and velocity profiles. For the continuity of the whole procedure, the. plotting process was included as a part of this program. The flow diagram is shown in Figure 25. BEGIN INPUT of incidence matrices and nodal coordinates in 3-D space and in selected cross-section, coeff. of A and B, and 3-D nodal pressure values l 89 DETERMINATION of the element number in 3-D space in which the node of the selected cross-section is located, and the nodal coordinates and the nodal pressure values related to this element l TRANSFORMATION of Cartesian coord. into natural coord. for the node in the selected cross-section by Newton-Raphson method j CALCULATION of the nodal pressure values and normal velocity and its components for the node in ' the selected cross-section l OUTPUT of the calculated pressure and velocity values i PLOT the pressure contour lines and the velocity profiles for the selected cross-section l END Figure 25. The flow chart of the computer program for the calculation of pressure and velocity in 2-D space 90 3.3.3 Calculation of streamlines and flow rate for the selected crow-section Streamlines for the selected cross-section can be determined through its definitions or through its properties. Thecalculationofflowrateisbasedonthefactthatthe flowrate will remain constant between two streamlines. This program also needs the incidence matrices and nodal pressure values in 3-D space as its input data. The output of this programwillbeusedtodraw streamlinesintwo—dimensionalspace, toobtainregression equations for the streamlines, and to calculate the flow rates. The flow chart of this program is shown in Figure 26. 91 BEGIN T INPUT of incidence matrices and nodal coordinates in 3-D space, coeff. A and B, and 3—D nodal pressure values l SELECTION of any point on the top free surface or in the duct opening of the selected cross-section to start the process for determining the streamline l DETERMINATION of the element number in 3-D space in which the node of the selected cross-section is located, and the nodal coordinates and nodal pressure values related to this element l TRANSFORMATION of Cartesian coord. into natural coord. for the node in the selected cross-section by Newton-Raphson method E CALCULATION of normal velocity and its components for the node in the selected cross-section l 92 DETERMINATION of next point to be traced by multiplying the velocity components by a time step i OUTPUT of the traced points which form the streamline l FORMULATION of the regression equation for each of the streamlines in the selected cross-section L CALCULATION of the flow rates between any two of the streamlines l OUTPUT of the flow rates l PLOT the streamlines and the sketch of the flow rate distribution l END Figure 26. The flow chart of the computer program for the calculation of streamline and flow rate in 2—D space CHAPTER4 ANALYSIS OF AIR FLOW PATTERNS 4.1 Preparation of basic data Torunthecomputerprograms, thevariablesaffectingairflowpattemsandtheir levelsweredecided,theparameters, suchasthedimensionandconfigurationofduct synemandpommstomge,meproperfiesofairmddwchamctuisficsofpomwmbers, were prepared, and themesh schemes were generated. Under present study, it was assumed that the air flow was inviscid, the air temperature was 10" C and the relative humidity was 95 96. 4.1.1 Basic data of duct system and potato storage The basic data for a duct system and potato storage were as follows. Storage dimension: L x W x H = 19.5 m x 9.1 m x 5.3 m Specific gravity of potato: g = 641.3 lrglnr3 Storage capacity: C, = L x W x H, x g/1,000 = 489.0 metric ton Requirement of air flow rate per metric ton: q = 0.94 m3/min/(metric ton) Totalairflowraterequired: Q = C, x q = 460.0m3 Air flow velocity in the main plenum: V, = 198.0 m/min 93 94 Cross-sectional area of the main plenum: A, = Q/V, = 2.3 nr2 Air flow velocity in the lateral duct: V, =- 259.0 m/min Air flow velocity exiting duct: V. = 305.0 m/min 4.1.2 Selection or variables and their lavels ' To evaluate the effects on the pressure and velocity distributions of the variables, such as duct size, duct spacing, lateral duct pressure, depth of potato pile, and the distance from a selected cross-section to the duct entrance, three levels for each of the variables and four duct shapes (triangular duct, circular duct, semicircular duct and rectangular duct) were selected as listed in Table 6. Table 6. Selectedvariablesandtheirlevels asusedinthecomputerprograms Lateral duct pressure Pd (Pa) Duct spacing 1.8 2.4 Lt (n!) 3.1 Depth of potato pile 3.1 4.3 HP ("1) 5.5 Triangular duct size 0.59 x 0.34 0.64 x 0.36 h X a. (m X m) 0.67 x 0.39 Circular duct size 0.51 0.54 d. (m) 0.58 Semicircular duct 0.36 0.38 r. (m) 0.41 Rectangular duct size 0.32 x 0.32 0.34 x 0.34 a, x b, (m x m) 0.36 x 0.36 Distance from selected cross- 2.0 6.0 section to duct entrance (m) 10.0 96 4.1.3 Coefficients in Shedd’s equation The granular permeability K0 is a function of coefficients A and B, which ean only be determined experimurtally. Staley and Watson (1961) reported A = 345 and B = 0.562 in English units for a range of pressure gradients (0.001 - 0.02 inch water/ft) and air flow velocities (6.0 - 50 ft’lminlftfi. Because the regression line of the experimental datainthelog-logscaleis usuallynotastraightline,AandBvaluesarenotconstant in a wide range of pressure gradients. Therefore, it is reasonable to use several segments of straight line instead of one single straight line to approach the test data as suggested by Segerlind (1983). To obtain coefficients A and B for apotato storage, Equation [15] and values ofa and b in ASAE Data Standard D272.2 (ASAE Standard, 1988) were first used to calculate the pressure gradient for a given air flow. Then, using a nonlinear regression method, various values of A and B (Table 7) were produced for the corresponding range of velocity and pressure gradient according to Fquation [l3]. 97 Table 7. Coefficients A and B of Equation [13] for air flow through potato storage. _- flow (m’ls/m’) gradient (Pal m) of A of B >0.05- 0.10 >1.5- 5.0 0.568 >0.10 - 0.20 >5.0 - 17.0 0.0411 0.558 >0.20 - 0.50 >17.0 - 90.0 0.0422 0.549 >0.50 - 1.00 ' >90.0 - 324.0 0.0435 0.542 >1.00 - 2.00 >324.0 - 1165.0 0.0446 0.538 4.1.4 Defining boundary conditions A sketch of the boundary conditions for a potato storage with triangular ducts is shown in Figure 27. Boundary conditions of the first type, Dirichlet conditions, are defined atthenodesonthesurfaceofGI-IIJ withP = 0, andonapartofsurfaceof ABCD with P = P... Boundary conditions of the second type, Neumann conditions, are defined at the nodes on the solid boundaries, AEHGB and DFIJC, on the symmetric boundaries, BGJC and EHIF, and on a part of surface of ABCD with dPldn = 0. The boundary conditions for potato storage with circular ducts, semicircular ducts or in-floor rectangular ducts are similar to those with triangular ducts. The loeations of the boundary nodes with P = P‘ for the above four duct shapes are also shown in Figure 27. ewes» 95oz e8 3029.8 bane-eon 3 e885 KN ”SEE use .285 [[[ll /\ a 'l’ 8.6 5:985. O cuemEm / / eueeee ohm 4.1.5 Calculation of duct presumes When air flows through a duct system, the pressure and velocity will change along the longitudinal dimension of the duct. The magnitude of the change depends on the cross-sectional area of duct, duct shape, roughness of duct interior surface, size and location of the duct openings, and the flow rate delivered by the duct. Thepressureandthevelocitychangesforaductcanbeobtained fromtheBemoulli equation (Sabersky et al. 1989): v22 P2 VI2 Pl _+_+ a: _+_+ +M-h [69] 2g pg Y2 Yr r where variables upstream and downstream are assigned by subscript l and 2, respectively. Each term in Equation [69] represents the energy per unit weight and has the dimension of length. The elevation heads, y, and y,, are measured at the center lines of the duct upstream and downstream, respectively. In the present problem they are equal. The function of pump work, M, is to increase the total head of the flow. While the friction loss, h,, is usually obtained from the friction factor: f = h' > 1 v2 [70] [.1 [2:] ? which is a function of Reynolds number and the relative roughness ratio, eJd, of the duct. The Reynolds number is defined as: 100 Rep . pV_d [71] p. where V. is the mean fluid velocity over the duct cross-section. Incropera and DeWitt, (1985) and Sabersky et al. (1989) cited the Moody diagram, as shown in Figure 28, to find friction factors for a wide range of Reynolds numbers. For fully developed laminar flow, the friction factor also can be ealculated from the following formula (Incropera and Dewitt, 1985): f-— [72] The Moody diagram (Figure 28) is designed for round ducts only. For noncircular ducts, such as triangular ducts, rectangular ducts and semicircular ducts, the friction factor also can be estimated from the Moody diagram by assigning the hydraulic diameter D, instead of d in the calculation of Reynolds number with D, = _° [73] where A, and P. are the flow cross-sectional area and the wetted perimeter, respectively (Incropera and DeWitt, 1985, and Sabersky et al. 1989). 101 S—i— ssatrrfinor utters}; $2 .3 a inseam Heeeem .3.» can. 865358 2528 macro.» we 8% 958 .8 e385: ago—Sum «o e385”. a an 286580 eceotm .mm 053m 88o... .... vac. 08. mos. 3.0 m 56 ~06 «.05 VD. and e we saw u a: con-es: 3.9.3: no. '0— moei Eco—em z j rows} uonopd ‘7") ( r.\_/ it vcd {W we a. em m. U - 3:: ._u:c. 2...... . 1.... 30: _t :. . 35:3.— 102 4.1.6 Mad: generation Meshgeneradonisanmjorstepinusingdwfifitedemartmethodmdiscrefizetwo- dimensional and three-dimensional domains. The accuracy and the cost of the solution processes of the finite element method depend largely on the mesh scheme employed. As mentioned above, a solid element with 20 nodes has been chosen for the three- dimensional problem. For the regions where the pressure or velocity may be expected to change dramatically, smaller elements should be used, otherwise larger elements were used. The generation of elements and nodes, if done by hand, is time consuming. There are automatic mesh generation programs (Segerlind, 1984) and some commercial software packages available to generate two- and three-dimensional elements and nodes (ANSYS Manual, 1987 and FIDAP Manual, 1989). The meshes generated by using FIDAP for a potato storage are shown in the Figure 29, 30, 31 and 32 for triangular duct, circular duct, semicircular duct and in-floor rectangular duct, respectively. 103 Figure 29. Mesh generation for potato storage with himng duct 104 h s? f Figure 30. Mesh generation for potato storage with circular duct 105 Figure 31. Mesh generation for potato storage with semicircular duct 107 4.2 'Iheeffectonairflowpatternsofvariablesunderstudy In the present study, it was found that for any selected cross-section, which is perpendiculartothecenterlineofthelatcralduct, thestreamlinesareparallelandthe airflowis uniformintheupperregion ofthepotatopilewhen thedepthofthepotato pile is equal to or greater than the spacing between two adjacent ducts, i.e. H, 2 L4. Nonlinear air flow usually dominates in the lower region of the potato storage where the effects of the variables under study become very distinct. Therefore, analysis of air flow patterns forthelowerpartofpotato storageappearsmoreattractivethanthatfordre upper part. In the following sections, all air flow patterns are shown only for the region that is under the depth of 2.1 m for the selected cross-section, and because of the symmetry, only half of the cross-section are displayed. Itwasalsonoted thatinthelowerpartofthepotatostorage, theareasnearthefloor and in the middle between two adjacent ducts are usually ventilated poorly. To highlight these areas, they were defined by a streamline, the symmetric line and the floor, and were shaded. Generally, in any given cross-section a lot of air streamlines can be drawn. But for the purpose of clarity and ease of comparison, the streamlines were chosen to be 0.24 m apart starting from the symmetric line at the top free surface of the potato pile and were progressing from the top free surface to the lateral duct. Since the velocity in the vicinity of the duct opening is larger than in other parts of the potato pile,- the time stepwassetinadecreasingpatterntoproduceanaccuratestreamline. In all Figures contained in section 4.2, the iso-pressure lines were represented by the percentage duct pressure and the velocity profiles were symbolized by their magnitude 108 inm/min.Thevolumetricairflowrate(L’t“),theairflowratepermetrictonofpotatoes (L’r‘M")ormassairflowrate(Mt")usuallyreferstoairflowrateinathree- dimensionalspace.Whileinatwo-dimensionalspacetheairflowrateperunitthickness (Lzr') was used. 4.2.1 The effect of duct spacing on air flow patterns To compare the effect of different duct spacings on air flow patterns, three duct spacings of 1.8 m, 2.4 m and 3.1 m were chosen. The related data and Figure numbers are listed in Table 8. Other variables and their levels were as follows: the duct pressure was 125 Pa, the depth of the potato pile was 4.3 m, and the dimension for the triangular duct wash, x a, = 0.59m x 0.34m, forthecircularductwasd, = 0.51 m, forthe semicircular duct was r, = 0.36 m and for the in-floor rectangular duct was a, x b, = 0.32 m x 0.32 m. The iso-pressure lines and streamlines for the triangular duct with duct spacings of 1.8 m, 2.4 m and 3.1 m are shown in Figures 33, 34 and 35, respectively. It is very clear from these Figures that there is a larger space between two iso-pressure lines in the lower right region than in the other parts of the cross-section. This means that the lower right region has a lower pressure gradient than in the other parts whether the duct spacing is large or small. In the region directly above the duct, the pressure distributions were fairly uniform for all three duct spacings. Comparing Figures 33, 34 and 35, it appears that the pressure value in the lower right region for Figure 33 is larger than in Figure 34, which in turn is larger than in Figure 35. 109 Inthepresentsmdytheflowisassumedtobesteadystate. Thustheairflowpathwill follow the streamline exactly. From Figures 33, 34 and 35. it can be observed that the streamlinesinthelowerregion ofpotatopilearenotparallel, sotheairflowis nonlinear. The path line directly above the duct is the shortest, while the path line near the symmetric line between two adjacent ducts is the longest. As a result of the unequal air flow path, it will take longer for the ventilation to bring the entire potato pile to the sametemperatumJtcanalsobesemdmtthesueamfinefornfingmeshadedaream Figure33 is shorterthaninFigure34, whichintumisshorterthaninFigure35. This shows that the air flow condition is dependent on the duct spacing. Potato storages with smaller duct spacing have more uniform air distribution than those with larger duct spacing. Suppose the air flow rates per metric ton of potatoes are the same for these three duct spacings, and consider the fact that in the two-dimensional domain the air flow rates in the area between two streamlines are constant, then the air flow per unit thickness in the shaded area can be obtained. The shaded area in Figures 33, 34 and 35 accounts for 32.4%, 27.0% and 24.2% of the total cross-sectional area, but only receive 26.7%, 20% and 16 % of the total flow rate per unit thickness, respectively. This indieates that the air ventilation is uneven through the selected cross-section with the shaded area poorly ventilated. The velocity profiles corresponding to the three different duct spacings are also shown in Figures 33, 34 and 35, respectively. They closely follow the relevant pressure distribution patterns. The velocities in the upper region of the cross-section are very uniform, while those in the region near the lower right comers are the lowest. 110 Theoretically, these corners are the stagnant points where there is no air movement at all beeause of the zero velocity. Decreasing the duct spacing is favorable for improving the aeration situation in the lower right part, which can be seen by comparing velocity profiles in Figures 33, 34 and 35. The discussion above is for triangular ducts. A similar tendency was observed for circular, semicircular and in-floor rectangular duets with the three duct spacings. The iso- pressure lines, streamlines and velocity profiles for these ducts are plotted in Figures 36, 37 and 38 for circular ducts, in Figures 39, 40 and 41 for semicircular ducts, and in Figures 42, 43 and 44 for in-floor rectangular ducts. The percentage shaded area over the total cross-sectional area and the percentage air flow rate per unit thickness of the shaded area for these three duct shapes with three duct spacings have been given in Table 8. 111 Table 8. The effect of duct spacing on air flow patterns: related data and Figure numbers Duct Shaded area Flow rate Figure No. referring Duct. spacing over total for shaded to iso-pressrue lines, shapes (m area (96) area (96) streamlines and velocity profiles , Triangular 1.8 32.4 26.7 Fig. 33 duct , Triangular 2.4 27.0 20.0 Fig. 34 duct Triangular 3.1 24.2 16.0 Fig. 35 duct Circular 1.8 33.1 26.7 Fig. 36 duct Circular 2.4 27.0 20.0 Fig. 37 duct . Circular ' 3.1 24.0 16.0 Fig. 38 duct Semicrrcul' ar 1.8 33.3 26.7 Fig. 39 duct Semicircular 2.4 27.0 20.0 Fig. 40 duct Semicircular 3.1 24.0 16.0 Fig. 41 duct Rectangular 1.8 31.5 26.7 Fig. 42 duct Rectangular 2.4 27.0 20.0 Fig. 43 duct Rectangular 3.1 24.3 16.0 Fig. 44 duct 31 112 a 3 e meeae use 55 .26 3.35.5 e2 23 852a .38.?» new Ace: 85.885 65 8:: §8&$_ .mm charm A8 $03 so coeeeuatx AE $000 so cozueétx 0.». gm 0; 0.0 QM ON 0; 0.0 \ , _ 0.0 e _ 0.0 h / $8 2 8 to; to; mu cm t_A $8 a .2 m. / -3 9 e V to.n W j *2. row. m; / I: m / , ,3. m V/ 10.0 106 w // toe 7/ roe A #8 0K / / ON (bU BVOQ'O X) UOnOauo-A 113 8 ed a 38% see .23 see same? 8. 25 85.3 bee—o.» 93 Eva 85.885 e5 3:: ogmoaefl em «are A8 $000 5 eozeeuatx A8 means Xv 8:085th 04. ohm ohm op. 0.0 . .04. ohm 0% on. 0.0 . / oo 00 L / g ro.— $8 10.— on 2 8 IA _ $2. ro.~ mu. roe 2 w / P 1:” w. V Tow / [I] J m/ g .58 roe .0 / roe Q 0 a... // IO.m 106 Max /” $8 toe % roe oK .// 89. ox (LU 87090 X) U0!139J!O-A 114 E $03 so coeeueotx 0.0 O.v _ O...” — E fin go 9.3an “one 55 8.6 5:983 e8 3th axe—mecca 3020.» 93 Eva mos—E85» e5 8:: 8385-9: .mm came A8 $03 XV coeeeeonx 0.0 L. S 0% o. — /. 2.. nu ma 8 0.0 I 0... rod #06 10.0 OK (Lu 817090 X) U0!139J!O-A on on 0% ow on odod s”... , see roe % s. L; % .